The Synchronized Dynamics of Complex Systems, Volume 6 (Monograph Series on Nonlinear Science and Complexity)

Contents Preface ix Chapter 1. Introduction 1 Chapter 2. Identical Systems 9 2.1. Complete synchronization 9 2...

Author: Stefano Boccaletti

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Contents

Preface

ix

Chapter 1. Introduction

1

Chapter 2. Identical Systems

9

2.1. Complete synchronization

9

2.2. The PC configuration

11

2.3. The APD configuration

14

2.4. Bidirectional coupling configurations

15

2.5. The stability of the synchronized motion

17

2.6. The master stability function 2.6.1 The case of time continuous systems 2.6.2 The master stability function for coupled maps

20 20 25

2.7. Assessing the synchronizability

25

Chapter 3. Nonidentical Systems

31

3.1. Phase synchronization of chaotic systems 3.1.1 Synchronization of periodic oscillators 3.1.2 Phase of chaotic signals 3.1.3 Phase entrainment of externally driven chaotic oscillators 3.1.4 Phase synchronization of coupled chaotic oscillators

33 33 36 41 45

3.2. Transition to phase synchronization of chaos

47

3.3. Experimental verification of the transition to phase synchronization

49

3.4. Ring intermittency at the transition to phase synchronization

53

3.5. Imperfect phase synchronization

58

3.6. Lag synchronization of chaotic oscillators

61

3.7. Intermittent lag synchronization

63

3.8. Phase synchronization of nonautonomous chaotic oscillators

69

3.9. Generalized synchronization

74

3.10. A mathematical definition of synchronization v

78

vi

Contents

3.11. Synchronization of structurally nonequivalent systems 3.11.1 From chaotic to periodic synchronized states

83 85

3.12. Systems with coexisting attractors

88

3.13. Methods and tools for detecting synchronized states 3.13.1 Detection of a functional relationship 3.13.2 Embedding and multivariate data 3.13.3 Interdependence between signals 3.13.4 Predictability of time series 3.13.5 Coupling direction 3.13.6 Detection of phase synchronization 3.13.7 Detection of local synchronization

94 95 96 98 99 101 101 104

Chapter 4. Effects of Noise

111

4.1. Noise-induced complete synchronization of identical chaotic oscillators

112

4.2. Noise induced phase synchronization of nonidentical chaotic systems 117 4.3. Noise enhanced phase synchronization in weakly coupled chaotic oscillators 4.4. Constructive noise effects in systems with noncoherent phase dynamics 4.4.1 Noise-induced changes in time scale and coherence resonance 4.4.2 Noise-induced complete synchronization 4.4.3 Noise-enhanced phase synchronization, deterministic and stochastic resonance Chapter 5. Distributed and Extended Systems

120 122 124 126 129 135

5.1. Synchronization in a chain of coupled circle maps

136

5.2. Phase synchronization phenomena in a chain of nonidentical phase coherent oscillators

140

5.3. Collective phase locked states in chains of phase coherent chaotic oscillators

142

5.4. In phase and anti-phase synchronization in chains of homoclinic oscillators

146

5.5. Synchronization domains and their competition

151

5.6. Synchronization in continuous extended systems

157

5.7. Asymmetric coupling effects

164

5.8. Defect enhanced anomaly in asymmetrically coupled spatially extended systems

173

Contents

5.9. Convective instabilities of synchronization in space distributed and extended systems Chapter 6. Complex Networks

vii

178 185

6.1. Definitions and measures of complex networks 6.1.1 Unweighted graphs 6.1.2 Weighted graphs 6.2. Coupling schemes in lattices 6.3. The master stability function

186 186 190 192 193

6.4. Key elements for the assessing of synchronizability 6.4.1 Coupling matrices with a real spectra 6.4.2 Numerical simulations 6.4.3 Coupling matrices with a complex spectra 6.5. Networks with degree–degree correlation

195 196 198 202 206

6.6. Synchronization in networks of phase oscillators 6.7. Synchronization in dynamical networks 6.8. Synchronization and modular structures

210 212 222

References

229

Subject Index

241

Preface

Cui dono lepidum novum libellum arida modo pumice expolitum? Corneli, tibi: namque tu solebas meas esse aliquid putare nugas, iam tum cum ausus es unus Italorum omne aevum tribus explicare chartis doctis, Iupiter, et laboriosis. Quare habe tibi quicquid hoc libelli, qualecumque quod, o patrona virgo, plus uno maneat perenne saeclo. Catullo, carme 11 There is an unwritten rule that states that a book should start with an epigraph. I frankly never understood if this is because a book has really to start this way. It is however a matter of fact that sometimes this observance flows into the choice of some apodeictic sentence, unravelling the meaning of which requires in most cases a great effort of the reader. Having to obey to such an usance, I had to think a lot for finding the proper way to start. I don’t like those ipse dixit sentences. Either, indeed, they have a clear meaning (and therefore one can express that concept without having to quote a sentence written by another), or they have a sort of fuzzy meaning, open to many different interpretations (and in this case they are, therefore, useless). To give an example, this is what I learned from reading a recent book by an Italian writer (Alessandro Baricco) where it is reported what a music critic (not at all a berk) wrote on the authoritative journal The Quarterly Musical Magazine and Review one year after the first representation of the Beethoven Ninth Symphony: 1 Whom I will dedicate to this new amusing booklet,

just cleaned by the dry pumice? To you, Cornelio, and indeed you were used to think that my “nugae” were worth something already when you dared, the only one among Italians, explaining all centuries within three papers, duct, Iupiter, and that costed you a lot of work. Therefore, have something of this booklet, whatever you want, so as, virgin protector, it could last perpetually for more than one century. ix

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Preface

“Elegance, pureness and moderation, that were the basic principles of our art, have progressively knuckled under a new style, flimsy and hasteful, that these times of airy abilities have adopted. Brains that, as for education and use, are unable to think to something else than clothes, fashion, gossip, novel reading and moral squandering, don’t get at tasting the more elaborated and less feverishness pleasures of science and art. Beethoven writes for those brains, and it seems that for this he is getting a rather big success, if I have to believe to the plaudits that, from all sites, flourish for his last work.” Now, I share fully the comment of Alessandro Baricco when he says that what makes one smile is precisely the fact that the Ninth Symphony today is considered a stronghold of traditional music against new tastes. That music began a flag, a hymn, a supreme underpinning of our civilization, and permeates our lives in unimaginable ways: on 1982, when Philips had to fix the standard for the size of a compact disk (about 12 cm radius), they decided that the new support had to be able to store in its entirety the best piece of music ever written, Beethoven’s Ninth Symphony. Sic transit gloria mundi! So, having discarded such a kind of incipit, I decided to use another option: to collect the various sentences that meant to me something during my life (not necessarily related to synchronization) and to start each chapter of the present book with one of them. And the most obvious choice for starting this Preface was the first carme of Catullo, a dedication for his collection of nugae, that accompanied my adolescence and my classic studies, and that genuinely reflects the genesis of the present book. During the last fifteen years, indeed, I had the chance of being part of a tight scientific community working on synchronization processes in nonlinear dynamical systems. I am greatly indebted to all members of such community for the friendly and warm atmosphere that characterized all our discussions about science, and that contributed to establish with any one of them not only a professional relationship, but also a profound fellowship, that will last regardless on the differences that will characterize our future scientific interests. As always happens in science, our ideas are questionable, whereas my deep debt of gratitude for their friendship is not. Ideally, this book is then dedicated to all my colleagues. Primarily, it comes to my mind Hector Mancini. It is a rare fortune for a man to have the chance to meet a such beautiful person, who always cared about me much beyond what the professional relationship would have required, always heedful to understand my needs, and glad to help. The time I spent with him at the Department of Physics and Applied Mathematics of the University of Navarra taught me what means having a friend, with whom sharing the good and the bad moments of the life.

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Together with him, I would like to mention the younger colleagues of the same Department, who shared with me the enthusiasm of my first studies on this subject: Diego Maza Ozcoidi, Javier Burguete, Wenceslao Gonzalez-Viñas, Jean Bragard. Our endless discussions in front of the white-board of the Department’s coffee room will last in my memory as one of the most inspiring moments of my career. I strongly hope to have left there, together with a good Italian Expresso coffee machine, also the attitude of realizing such a brain storming processes that were so helpful and important to delineate joint activities and to inspire new research lines. Immediately after, I would like to thank gratefully those colleagues with whom, some time after, I shared the effort of writing a long monographic review manuscript on synchronization, published on Physics Reports. I feel specially indebted to Jüergen Kurths, Diego Valladares, Grigory Osipov and Chansong Zhou. The present book is an extension of that review paper, and contains a lot of the material we collected and discussed together. To all of them goes my deepest gratitude for the so many scientific discussions we shared, and for the so much of the present subject I learned from them. And how to forget the many other colleagues who have accompanied my studies during the last years, and with whom I shared discussions, comments, remarks and other good and funny moments: Kenneth Showalter who always supported my research; Lou Pecora, a real genius and a real gentleman; Arkady Pikovski who, all the times we meet, addresses me in a perfect Italian: “Tutto bene?” (“Is everything OK?”); Michael Rosenblum, Michael Zaks, Vadim Anishchenko and the many vodka’s that accompanied our meetings; Rajarshi Roy who always likes to show pictures of mine in his excellent talks; my great Israeli friends Itamar Procaccia and Eshel ben Jacob who helped me in so many circumstances, Eckehard Schöll, Celso Grebogi and the great moments we had together; Ulrich Parlitz, Eric Kostelich, Vito Latora, Gabriel Mindlin, Maxi San Miguel, Vicente Perez Muñuzuri, Regino Criado, Mike Shlesinger. And the greatest young scientists I had the chance of meeting in Spain, such as Yamir Moreno Vega, Jordi Garcia Ojalvo, Irene Sendiña Nadal, Inmaculada Leyva, Ines Perez Mariño, Javier Buldú who contributed with so many beautiful ideas to my reasonings and thinking. Together with my advisor Fortunato Tito Arecchi, it is my pleasure to mention all the other colleagues with whom I collaborated on issues related to the subject of synchronization: PierLuigi Ramazza, Stefania Residori, Gianni Giacomelli, GianPiero Puccioni, Riccardo Meucci, Marco Ciofini, Antonino Giaquinta, Alessandro Farini, Fréderic Plaza, Manuel Velarde, Antonino Labate, Roberto Genesio, Livio Narici, Silvia Soria, Ying Chen Lai, Enrico Allaria, Maria Luisa Ramón, Santiago Madruga, Manuel Matías, Andrea Vallone, Antonio Pelaez, Sergio Casado, Umberto Bortolozzo, Tom Carroll, Fred Feudel, Ricardo López Ruiz, Murilo Baptista, Luc Pastur, Kresimir Josic, Alexander Hramov,

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Alexey Koronovskii, Andreas Amann, Francesco Sorrentino, Mario Di Bernardo, George Hentschel, Alexander Pisarchik, Mark Spano, Bruce Gluckmann, Luigi Fortuna, Mattia Frasca. I definitely have to recognize that my personal contribution to the field stands much below what I could receive from being part of such a wonderful scientific team, in terms of the many human friendly relationships I had the chance of establishing with these people and with the many others that unavoidably I’ve forgotten to mention. I remember with particular gratefulness all my students, from the first one Santiago de San Román (and our endless card games), to the last one Dong-Uk Hwang, to all the others: Carolina Mendoza, Mario Chavez, Italo Bove, Guillermo Huérta Cuellar. In most cases, they are now established researchers, and I feel particularly indebted to them for the innumerable inspirations and ideas we had the chance of sharing, as well as for all those unrepeatable moments that characterized our collaboration. My fond thought goes, also, to two other colleagues who are not anymore with us, and whose absence we all feel: Lorenz Kramer and Carlos Pérez García. With the first I had the chance of interacting when I was yet in my early stage of the career, reaching to appreciate his incredible knowledge of nonlinear science, together with his friendly attitude in sharing his mastery on the subject with the younger colleagues. With the second I passed a much longer time when we both were Professors in the same Department of Physics in Spain, learning so much about my professional activity. Furthermore, this book is dedicated to all persons who made (and make) it possible for me to follow my passion for science. I will never forget two of my Professors whom I had the chance of having during my classic studies at the end of the eighties: Profs. Chiara Asselle and Paolo Chiarelli. Prof. Chiara Asselle was my professor of Latin and ancient Greek at that time. Still now, more than 20 years after I finished that part of my education, my dreams are populated by her fantastic classes of Latin and Greek literature. Definitely, my way of getting involved in research, and of getting curious about discovering what is beyond reality, is so influenced by what I could learn from her passionate explanations of the philosophy and lyric developed in those ancient years. Prof. Paolo Chiarelli was my professor of mathematics and physics in those early years, and he is actually the main responsible for my final preference for scientific studies in the University. It was a real fortune to have him as a professor during those years, as he really initiated me to the beauty of physics, and transmitted me the passion for research, as well as the attitude of scientifically investigating the world around me. Moreover, I would like now to express my gratitude to the Italian Ambassador in Israel His Excellence Sandro de Bernardin, to the Director of the Italian Cul-

Preface

xiii

tural Institute Simonetta della Seta, to the Councellor of the Embassy Davide La Cecilia, to the commercial Councellor Tiziana D’Angelo, and to all the personnel of the Italian Embassy in Tel Aviv, with whom I will share the pleasure of working during the next future. And finally, I want to thank most gratefully all the members of my family: my mother Laura and father Mario, my wife Cristina and my daughter Chiara. They always supported my activities in the strongest as possible terms, helping me in all required circumstances. I am totally aware that my entire scientific career couldn’t have developed without their continuous bearing. But now, it is time to start.

Chapter 1

Introduction

The larger is chaos the larger the option for a man to control the natural phenomena. It was a great sunny day, that 10th of June 2000 in London. Thousands and thousands of citizens, the City’s Authorities, the most of the Press and local and international broadcasting channels were attracted to participate to a special event: the Opening Ceremony of the London Millennium Bridge. But, nobody even imagined that they would have instead assisted to one of the most spectacular manifestations of what a nonlinear synchronization phenomenon is. As soon as the many people, indeed, put over the bridge and started to walk, something suddenly happened: the bridge started to wobble from side to side! The following is the report of the event that is still appearing in the BBC News Site:1 “Some 80,000 people crossed the bridge on its opening day and those on the southern and central spans detected vibrations. The bridge began to sway and twist in regular oscillations. The worst of the movement occurred on the central span where the deck was moving by up to 70 mm. The frequency of the oscillations increased, leaving people unnerved and unsteady. The engineers insisted the bridge wouldn’t fall down but closed it completely after an attempt to limit numbers proved unworkable. Engineers discovered that the sideways forces of the pedestrians’ footsteps created a slight horizontal wobble in the bridge. As the structure began moving, pedestrians adjusted their gait to the same lateral rhythm as the bridge. The adjusted footsteps magnified the motion—just like when four people all stand up in a small boat at the same time. As more pedestrians locked into the same rhythm, the increasing oscillations led to the dramatic swaying captured on film. The sideways motion has been seen before, most notably in 1975 on the Auckland Harbour Bridge in New Zealand.” 1 http://news.bbc.co.uk/hi/english/static/in_depth/uk/2000/millennium_bridge

1

2


A movie is also found in the same Web site, that witnesses the dramatic locking effect of the pedestrian footsteps, whose synchronization is the cause of the observed Millennium Bridge’s motion. Though in that occasion the wonder and surprise of both experts and citizens was big, the understanding of how collective (synchronized) phenomena set up in the evolution of coupled dynamical systems goes back historically up to the earlier days of physics. Already in 1665, indeed, Christiaan Huygens introduced the novel concept of “sympathy” observing the collective (synchronized) rhythm of two pendulum clocks suspended by the side of each other, that swung with exactly the same frequency and in a 180 degrees out of phase configuration [1]. Huygens further noticed that such an anti-phase synchronous state was robust against perturbations (if disturbing the pendulums, the “sympathy” state was eventually restored and persisted indefinitely), and deduced that the fundamental cause of this effect was the very tiny coupling coming from the imperceptible movements of the common frame supporting the two pendulums. Furthermore, if one carefully observes natural phenomena, one immediately realizes that synchronous behavior of slightly interacting dynamical units is an ubiquitous phenomenon and permeates many aspects of our daily life: from the spectacular synchronization of the blinking behavior of fireflies of certain species that one can witness in some trees on the side of rivers in South-East Asia, to the synchronization in the calling behavior of Japanese rain frogs (whose origin is the attempt of the males of those species to better attract females for mating). And, again, the perfect synchrony of the rotation and revolution periods of the Moon which is the cause of the fact that we always observe the same face of the Moon from the Earth, or the studies on women who spend a lot of time together and show evidence of developing menstrual synchrony, in the sense that their interaction can lead them to menstruate around the same day of the month. Further examples are the interaction of vibrations of some adjacent organ pipes, which is known to cause, in certain occasions, a perfect unison or can even have the dramatic effects of reducing the sound of the pipes to almost silence, and the evidence that some normal and abnormal behavior of the human brain (including some brain diseases) are the result of a sudden and abrupt synchronization in the activity of a large number of neuronal populations. There was even who imagined that a time synchronized jumping of millions of persons could have caused a permanent deviation in the revolution trajectory of the Earth around the Sun, leading to drive the planet into a new orbit: the web-site http://www.worldjumpday.org reports the results of a very funny social experiment, consisting in synchronizing the jump of millions of people on July the 20th, 2006 (the World Jump Day) at 11.39.13 GMT. The same colloquial meaning for the word synchronization has its very old root in the ancient Greek. Synchronization comes, indeed, from the Greek σ υγ `


3

χρ oνoς ´ which means “to share the common time”, and this original meaning has been maintained up to now as agreement or correlation in time of different processes [2]. The scientific definition of synchronization refers, instead, to a general process wherein two (or many) dynamical systems (either equivalent or nonequivalent) are coupled or forced (periodically or noisy), in order to realize a collective or synchronous behavior. In particular, when one deals with nonlinear chaotic or complex dynamical units, the arousal of such collective (synchronized) dynamics is, in general, far from being trivial. Indeed, a nonlinear dynamical unit is said to produce a chaotic motion when its evolution depends crucially on the initial conditions. This implies that even two identical (but separated or uncoupled) systems that would evolve from almost identical (i.e., only slightly differing) initial states, would give rise to two trajectories that would naturally and exponentially separate in time. As a consequence, one could say that chaotic systems are systems that intrinsically defy synchronization, as even when the systems are identical but start from slightly different initial conditions, the corresponding trajectories would evolve in time in a unsynchronized manner. This relevant practical problem (in experiments or in nature one generically has initial states that are never known with infinite precision) started a great interest on the study of how a collective (synchronized) behavior can be set in coupled chaotic systems. When one regards synchronization phenomena in coupled chaotic systems, one can assume at least three distinct points of view for classifying the different observed behaviors. The first way is to classify the observed collective states depending on the nature of the synchronization phenomena. And, indeed, in the context of coupled chaotic elements, many different synchronization states have been studied in the past years. They are: complete or identical synchronization (CS) [3–5], phase (PS) [6,7] and lag (LS) synchronization [8], generalized synchronization (GS) [9,10], intermittent lag synchronization (ILS) [8,11], imperfect phase synchronization (IPS) [12], and almost synchronization (AS) [13]. CS was the first discovered phenomenon and it can be considered as the simplest form of synchronization. Its emergence consists in a perfect hooking of the chaotic trajectories of two coupled identical systems, in such a way that they remain in step with each other in the course of the time [5]. This phenomenon strictly corresponds to the case for which the synchronization error (the difference between the states of the two coupled systems) vanishes asymptotically. GS is instead a more complicated phenomenon which involves the coupling of nonidentical systems, and whose arousal corresponds to the emergence of a

4


function (not necessarily the identity) that associates the output of one system to that of the other system [9,10]. As this function might well be not invertible, one immediately realizes that the concept of GS is an asymmetric concept, in that one can have situations where the state of one system can be predicted by measurements of the other system, but not vice versa. An intermediate regime occurring in coupled nonidentical system is phase synchronization (PS), wherein a locking of properly defined phases is produced, without implying in general a corresponding high correlation in the amplitudes of the systems [6]. PS corresponds, therefore, to a weaker stage of synchronization, which generally occurs already at very small coupling strengths, if compared with the values needed to generate CS or GS. LS implies the boundedness of the difference between the actual output of one system and the delayed output of the other. The shift in time τlag making the equivalence of the two systems is called lag time [8]. In its turn, it is evident that this phenomenon implies that the two outputs lock both their phases and amplitudes, but with the presence of a time lag [8]. ILS is a state in which the two systems verify LS for most of the time, but intermittent bursts of nonsynchronous behavior occur persistently [8,11], generally in correspondence with the passage of the system’s trajectory in particular attractor regions where the LS state is locally unstable [8,11]. In analogy with ILS, IPS is a regime where phase slips (2π jumps in the phase difference of the two systems) occur persistently within a PS regime [12]. This synchronization regime is characteristic of coupled oscillators that are not phase coherent, i.e., such that they oscillate at a wide distribution of frequencies. Finally, AS results in the asymptotic boundedness of the difference between a subset of the variables of one system and the corresponding subset of variables of the other system [13], without implying a specific kind of relationships between the variables not belonging to the two subsets. Given a pair of coupled systems, one can observe a scenario of transitions among different types of synchronization. The first observation of such transitions was described for symmetrically coupled nonidentical systems and consisted in successive transitions between PS, LS and a regime similar to CS when increasing the strength of the coupling [8]. The second point of view is to distinguish different synchronization phenomena as a function of the nature of the coupling. For instance, there is a great difference in the process leading to synchronized states, depending upon whether the coupling is symmetrical or asymmetrical. In particular, one should distinguish two main cases: the unidirectional and the bidirectional coupling configuration. In the former case, there is a drive (or master) system whose output influences the behavior of a response (or slave) system. This implies that one system (the master) evolves uncoupled and drives the evolution of the other (the slave).


5

A very different situation is the one described by a bidirectional coupling, where both subsystems are coupled to each other, and the coupling factor induces an adjustment of the dynamics onto a common synchronized manifold. Of course one can think of any intermediate stage of asymmetrical coupling, i.e., all possible configurations where both systems influence one another’s trajectories, but with different strengths. Furthermore, one has to consider that the coupling nature has not necessarily to be linear, and, indeed, many relevant situations exist that are better described by a nonlinear coupling, or an impulsive coupling (i.e., a coupling that acts only at discrete time intervals). As one can expect, the nature of the coupling strongly influences the final synchronized state. For instance, GS is believed to be intimately related to a unidirectional coupling configuration, whereas the very same features of CS states can be quantitatively related to the degree of asymmetry that is present in the coupling scheme. Furthermore, up to now no way has been proposed to reduce bidirectional effects to unidirectional processes, nor to link formally the two cases. The third point of view, that will be adopted along the presentation of this book, is to distinguish the different cases of synchronization as a function of the nature of the coupled systems. This book starts, indeed, with the presentation of the case of two (or many) coupled identical systems. Precisely, in Chapter 2 we describe the first historical studies on CS phenomena of coupled chaotic systems, both for low and for high dimensional situations, and discuss the relevant problem of assessing the stability conditions for the synchronization manifold. In this context, we report a very nice analysis (the so-called Master Stability Function approach) that gives rigorously the necessary conditions for a synchronization state to appear in a stable way for an arbitrarily coupled set of identical dynamical units. Chapter 3 moves to the case of nonidentical systems, and summarizes the novel synchronization regimes that arise in this context. A particular emphasis is given to the transitional behaviors from nonsynchronized to synchronized states. The chapter ends with the trial of an unifying framework that encompasses within a single definition the different observed synchronization phenomena, and with the description of the main methods and tools that can be used for detecting synchronization from multivariate data sets. Chapter 3 also extends the analysis to the case of structurally different systems. Here, two main situations are considered: the case of two coupled units such that their uncoupled dynamics are confined onto chaotic attractors with different structural properties, and the case of two unidirectionally coupled multi-stable units in which the states of the drive and response systems are initially prepared in two different coexisting attractors. Chapter 4 describes a situation that is frequently encountered in real system. Precisely, it is considered the case of uncoupled systems (or very weakly cou-

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pled systems) that would not give rise to synchronized motion, and it is discussed how an additional source of noise can induce or enhance the synchronization features. It has to be remarked that such counterintuitive and constructive effects are peculiar of the interplay between a noisy source and nonlinear systems, and therefore are of great interest in the analysis of real systems where the presence of additional noise is unavoidable. To make an example, the chapter ends with describing a real experimental situation where all these effects are clearly visible and can be proved in controlled laboratory conditions. In Chapter 5 we discuss the case of distributed and extended systems. The chapter starts with the description of large ensembles of chaotic units, and then makes an overview of synchronization effects in coupled continuous space-extended systems, i.e., systems extended in space whose evolution is ruled by partial differential equations. In this latter context, a particular attention will be devoted to describe the peculiar effects induced by controlled asymmetries in the coupling, that produce and enhance anomalous transitions to phase synchronized states. Finally the chapter discusses the problem of stability of synchronization in spatially distributed or extended systems, and shows that, in addition to the criteria of absolute stability, new instabilities, intimately related to the spatially extended nature of the systems (as, e.g., convective instabilities), can affect synchronization, leading to its destruction also in the cases in which the conditions for its absolute stability are verified. The final chapter is dedicated to describe the synchronization processes emerging in complex networks of dynamical units, i.e., in large ensembles of chaotic systems that are coupled by means of a wiring of connections that is featuring complex topological properties. After a small introduction to the definitions and measure of complex networks, the chapter discusses the Master Stability Function Approach in complex graphs, and shows how one can properly weight the connections of a network to greatly enhance its propensity to give rise to a synchronous behavior. The chapter ends with a survey of synchronization effects in dynamical networks, i.e., in networks where the wiring of connection is itself evolving in time, and with the description of a relevant application of synchronization to detect and identify the hierarchical structure of modules observed in most of the real world networks. At the end of this introductory chapter, it is mandatory to mention that a number of review articles and books on synchronization of complex dynamics have appeared in recent years, witnessing the interest and the flurry of researches that permeates the scientific community worldwide on this subject, and that the reader may find useful to consult. In particular, I would like to mention the book written by Arkady Pikovski, Michael Rosenblum and Jürgen Kurths entitled “Synchronization. A Universal Concept in Nonlinear Systems” [14] and published by Cambridge University Press in 2001, that contains a very nice account on the subject of synchroniza-


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tion in both periodic and chaotic nonlinear systems, and the monographic article entitled “The synchronization of chaotic systems” [15] published one year later in Physics Reports, which centers the attention on many issues whose treatment will be extended in the present book. It has to be mentioned, furthermore, that there is also a very nice book written by Steven Strogatz entitled “Sync: The Emerging Science of Spontaneous Order” [16] and published in 2003 by Hyperion Press, that succeeds in presenting the main concepts of synchronization in a very popular way also for a lay audience and readership.

Chapter 2

Identical Systems

S’ io fosse foco, ardarei al mondo, s’io fosse vento, lo tempestarei, s’io fosse acqua, i’ l’annegherei, s’io fosse Dio, mannerei’l nel profondo. S’io fosse Papa, allor sare’ giocondo, che’ tutti li gristiani imbrigherei, s’io fosse Imperator, sa’ che farei, a tutti mozzerei lo capo a tondo. S’io fosse morte, andaria da mi padre, s’io fosse vita, fuggire’ da lui, similemente faria da mi madre, s’io fosse Cecco, come sono e fui, torrei le donne giovani e leggiadre le vecchie e laide lasserei altrui. Cecco Angiolieri1

2.1. Complete synchronization As we discussed in the Introduction, complex systems (such as, e.g., chaotic systems) are dynamical entities that defy intrinsically synchronization, because of their essential property of displaying a crucial sensitivity to differences in the initial conditions. Therefore, the problem of assessing the conditions by means of which identical coupled systems set into a synchronized motion (thus loosing memory of their difference in the initial states) is far from being a trivial issue, and the research 1 If I was fire, I’d burn the world, / if I was wind, I’d storm it, / if I was water, I’d flood it, / if I was God, I’d send it in the deep Hell. / If I was Pope, then I’d be happy, / cause all Christians I’d cheat, / if I was Emperor, do you know what I’d do, / I’d cut the heads to everybody. / If I was Death, I’d go to my father, / if I was Life, I’d get out of him, / the same I’d do with my mother. If I was Cecco, like I am and have been, / I’d keep the young and beautiful women with me, / while the old and dirty I’d leave to the others.

9

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has historically started with the problem of synchronizing identical complex units, so as to make them evolving on the same chaotic trajectory [3–5,17,18]. When one deals with identical systems, the specific synchronized motion that emerges is complete synchronization (CS), the most natural synchronization state corresponding to the equality of the state variables of the two systems while they evolve in time. In a geometric representation, CS corresponds to the collapse of the full system’s trajectory in the phase space onto an identity hyperplane (the so-called synchronization manifold). As a consequence of this process, generally the appearance of CS is accompanied with an effective abrupt decrease of the dimensionality (complexity) of the overall motion. On its turn, this reduction in dimensionality is responsible for the fact that often synchronization processes are regarded as the spontaneous emergence of a certain degree of order within a complex dynamics. The phenomenon of CS was named differently in the literature. Other names that were used for indicating a similar behavior are Identical Synchronization, or Conventional Synchronization [19]. The first sections of this chapter are dedicated to describe and discuss the main properties of this synchronized motion. In doing that, we will follow the historical course of the research on this topic, and the discussion will be mainly concentrated on time continuous systems (such as systems described by sets of ordinary differential equations). However, most of the ideas that will be described apply directly (or can be easily extended) to discrete time systems as well (i.e., to chaotic mappings). It is essential to distinguish between two different cases, that differ intrinsically for the nature of the coupling scheme adopted to couple the dynamical units. One possibility is, indeed, to adopt an unidirectional coupling, often referred to also as drive–response or master–slave coupling configuration. This situation corresponds to the case in which the evolution of one of the coupled systems (the master) is unaltered by the coupling, while that of the other (the slave) is directly forced by the evolution of the master. The other possibility is that of a bidirectional coupling, in which the dynamical units are mutually coupled and influenced by each other’s behavior. This classification will serve us to describe the appearance of synchronous motion in identical systems. In fact, as we will see in the following, when the considered units are made different, these two schemes do not exhaust the description of the possible coupling configurations, and one has to explicitly study the full spectrum of asymmetrical configurations in the coupling, i.e., all possible situations where the influence of a system onto the other is explicitly and controllably taken larger in one of the two directions of the coupling. It is evident that the unidirectional and bidirectional cases represent the limits of a fully asymmetrical, and fully symmetrical coupling, respectively.


11

Historically, complete synchronization and the assessment of the robustness of the complete synchronization states was first investigated in a series of different coupling schemes, among which we recall the Pecora and Carroll method [5,18, 20], the negative feedback scheme [21], the sporadic driving configuration [22], the active-passive decomposition [23,24], the diffusive coupling, and other hybrid methods [25]. Furthermore, Ref. [26] made the first attempt of comparatively analyzing some of these different coupling configurations, and tried to encompass them within a rigorous mathematical framework. For the sake of illustration, the following sections will be mostly focussed on the essential points of some particular coupling schemes. Namely, the next two sections will summarize the details of the Pecora and Carroll (PC) configuration and of the Active–Passive decomposition (APD) method, with a particular emphasis given to the assessment of the stability of the synchronized motion. Later on, we will move to the case of an ensemble of arbitrarily (but bidirectionally) coupled identical chaotic systems, and we will describe the full details of the so-called Master Stability Function approach, that allows for a rigorous treatment of the stability in the proximity of the synchronization manifold, and that gives the necessary conditions for the complete synchronization manifold to be stable against infinitesimal perturbations.

2.2. The PC configuration Let us start then with discussing the PC configuration. In order to proceed, we initially consider a single chaotic system whose temporal evolution is ruled by z˙ = F (z).

(2.1)

In equation (2.1), z ≡ {z1 , z2 , . . . , zn } is an n-dimensional vector, whose components are the n components of the vector state of the system in the phase space, and F : Rn → Rn is the function that rules the evolution of the chaotic trajectory. The PC synchronization scheme makes the explicit assumption that equation (2.1) be drive decomposable. This means that the original n-dimensional system can be split into three subsystems, described by u˙ = f (u, v) driver, (2.2) v˙ = g(u, v) ˙ = h(u, w) response w where u ≡ {u1 , u2 , . . . , um }, v ≡ {v1 , v2 , . . . , vk }, w ≡ {w1 , w2 , . . . , wl }, and n = m + k + l. The subsystem of equations (2.2) that includes the vectors (u, v), defines the drive system, whereas the other subsystem of equations (2.2) represents the response system. Indeed, it can be easily seen that the evolution of the vector w is guided by the driver trajectory by means of the driving vector u.


12

In this scheme, one defines complete or identical synchronization as the identical evolution of the trajectories of the response system and of one replica of it ˙ = h(u, w ), w that is subjected to the same chaotic driving signal (u(t)). In other words, by defining synchronization error the norm of the difference between the response system and its replica e ≡ w − w , the arousal of complete synchronization implies that lim e = 0

t→∞

when using the same driving signal, or, in other words, that the response system is asymptotically stable. The condition that the synchronization error would vanish asymptotically is tantamount to say that the response system looses the memory of its initial conditions, i.e., different replicas starting from different initial conditions will converge to the same asymptotic trajectory in the phase space when subjected to the same driving force. It is important to remark that this latter statement does not imply any constrain on the final asymptotic state, which, in fact, corresponds to a trajectory evolving on top of a chaotic attractor. The important point that was established since the first researches on this configuration (Refs. [5,18]) is that such a synchronization process (and the consequent vanishing of the synchronization error) can be achieved only if all the Lyapunov exponents of the response system conditioned to the action of the driver (and therefore called Conditional Lyapunov exponents), are negative. In this case, u is termed as a synchronizing signal, and it is straightforward to demonstrate that not all possible decompositions of equation (2.1) lead to a synchronized state in the response system, since not all possible set of sub-variables can form a synchronizing signal. We now move to illustrate an example of such a coupling scheme, by building a PC configuration using as the drive the Lorenz system and as response the subspace of the Lorenz system containing the (x, z) variables [27] ⎫ x˙ = σ (y − x) ⎬ y˙ = −xz + rx − y driver, (2.3) ⎭ z˙ = xy − bz y˙ = −xz + rx − y response. z˙ = xy − bz


13

Figure 2.1. The complete synchronization regime of equations (2.3). Dashed (solid) line represents the temporal evolution of the variable z (z ). Notice that, after a transient, the two evolutions collapse into a single line, indicating that the two variables z and z become synchronized.

If one sets the parameters to be σ = 16, r = 45.92 and b = 4, equations (2.3) give rise to a chaotic dynamics. In addition, it has to be mentioned that the specific configuration described in the example of equations (2.3) is known as homogeneous driving configuration. This name comes from the fact that h ≡ g. With this particular choice of the driving, the complete synchronization state can arise in the system, as it is illustrated in Figure 2.1, where one sees that the temporal evolutions of the variables z and z (that constitute a subset of the response system and of its replica) asymptotically converge to the same value. This is not always the case for the presented example. Indeed, if one splits the main Lorenz system of equation (2.3) in a different way, not always a synchronizing signal can be defined. Precisely, the necessary condition for synchronization (the negativity of the conditional Lyapunov exponents) is met only for two choices, namely (x, z) driven by y and (y, z) driven by x. The third possible choice ((x, y) driven by z) gives on the contrary a positive conditional Lyapunov exponent, and therefore it leads to an instability of the CS dynamics. A more detailed discussion on the CS features of the Lorenz system can be found in Ref. [28]. Though the PC configuration is not the most general way to couple identical dynamical systems (as it requires divisibility of the original system), such a scheme, together with the one considered in Ref. [4], were historically the first proposed configurations with the main aim to show explicitly chaotic synchronization as a new and important feature emerging in coupled systems, while other


14

previous or contemporary studies had surfaced this idea in the analysis of arrays of coupled systems [3,29]. In the following section, we will continue our discussion and describe an alternative and more general configuration for a drive–response scheme, called the Active–Passive decomposition method.

2.3. The APD configuration This particular scheme was introduced in Ref. [23] a few years after the introduction of the PC configuration. It constitutes a more general driver–response scheme, in the sense that a larger class of systems can be encompassed by such a coupling configuration. The method was called the Active–Passive decomposition method (APD), and consists in explicitly considering the chaotic autonomous system of equation (2.1) as decomposed into the sum of a nonautonomous system (the response system) x˙ = f x, s(t) (2.4) and of an autonomous system (the driving one) given by s = h(x) or s˙ = h(x). The condition for the arousal of CS here is the very same as that of the PC configuration, i.e., the asymptotic identity between the evolutions of the response system (2.4) and one replica of it, when subjected to the same driving signal. In its turn, this implies that such a condition can be fulfilled only if equation (2.4) is asymptotically stable. Once again, one has to remark that this latter statement does not exclude the chaotic behavior of x(t), since its evolution is driven by the chaotic signal s(t). An example can be made for illustrating the APD configuration. We take the very same coupling scheme considered in the original Ref. [23] for the Lorenz system: x˙ = −10x + s(t), y˙ = 28x − y − xz, z˙ = xy − 2.666z with a driving signal constituted by s(t) = h(x) = 10y.

(2.5)

2.4. Bidirectional coupling configurations

15

In these conditions, the response and one replica of it are able to synchronize, but, by making use of a Lyapunov Function approach, Ref. [23] demonstrated that the response system can synchronize with its copy also for several other types of driving signal s(t), thus proving the generality of the method. Indeed, we have already mentioned that the PC scheme described in the previous section allows a given chaotic system to synchronize only for a finite number of possible decompositions. In the present case, instead, the freedom to choose the driving signal s(t) (or alternately the function h(x)) makes the APD scheme very powerful, general, and extremely flexible for being applied in various different circumstances. Obviously, the set of possible unidirectional coupling configurations cannot be fully encompassed in the two cases discussed before, and in fact several other drive–response schemes have been considered, many of them explicitly devised for application to communication by means of a cascade of chaotic system [24, 25,30–32]. A further consideration is in turn, and concerns the fact that some proper unidirectional configurations between identical systems may even induce synchronization phenomena that are different from CS. An interesting case is the so-called “anticipated synchronization” [33], a phenomenon occurring for different unidirectional coupling schemes of identical systems that include a time-delayed feedback either in the driver or in both coupled systems. These schemes may induce a synchronization state where the variables of the response system “anticipate” in time the values that will be assumed by the variables of the driver system (i.e., the response system synchronizes completely and identically with a future state of the driver system). The results of Ref. [33] elucidate that the anticipating synchronization manifold (where the response anticipates the driver) can be made globally stable due to the interplay between delayed feedback and dissipation, for any value of the lag time between response and driver. An example of this synchronization state is provided in Ref. [33] with reference to a pair of coupled Rössler systems [34], where a time-delayed term is introduced in the dissipative coupling. Later on, we will return to this specific synchronization process, to show that, once one implements the scheme on a chain of coupled identical systems, some peculiar spatial instabilities may arise also in the case in which the simple absolute stability of the synchronization manifold is warranted.

2.4. Bidirectional coupling configurations Up to now, we have mainly focused our discussion on unidirectional coupling configurations, in which one chaotic system (the master) is unaffected by the


16

coupling, while the other is driven by a suitable function of the master’s trajectory. Here, instead, we briefly discuss the case of a bidirectional coupling scheme, where both systems are mutually coupled. It is intuitive to understand that unidirectional and bidirectional coupling configurations differ substantially for what concern the processes leading to synchronized states. In the unidirectional case, the global system can be always seen as decomposable into two or many subsystems, that realize a drive–response (or master–slave) configuration, implying that one subsystem evolves uncoupled and drives the evolution of the other. As a result, the response system is slaved to follow the dynamics (or a proper function of the dynamics) of the drive system, which, instead, purely acts as an external forcing for the response system. In the bidirectional case, instead, both subsystems are coupled to each other, and the coupling factor induces an adjustment of the dynamics onto a common synchronized behavior. In general, a linear bidirectional coupling between identical chaotic systems can be considered as the introduction of an additional dissipative term in the dynamics: · (y − x)T , x˙ = f (x) + C · (x − y)T . y˙ = f (y) + C

(2.6)

Here x and y are the N-dimensional state vectors of the chaotic systems, while f C is an n×n matrix whose coefficients is a vector field f : Rn → Rn . Furthermore, rule the dissipative coupling, and the symbol T stands for matrix transposition. As one increases the coefficients of C (which is tantamount to strengthen the coupling between the two systems), system (2.6) displays a transition to a CS state at a critical value of the coupling. This critical value of the coupling depends on the particular structure of the coupling matrix. Here, the CS state corresponds to the asymptotic condition lim |x − y| = 0.

t→∞

A simple case that can be straightforwardly treated is the one corresponding to C = c I, i.e., when the coupling matrix is a multiple of the identity matrix. Here, the systems synchronize in a complete way for c > λL , where λL is the largest Lyapunov exponent of the uncoupled chaotic system. This result, discovered in Ref. [3], has a simple explanation. The reason for this transition, indeed, is that the long term behavior of the coupled systems can be seen as determined by two counterbalancing strengths: from one side there is the action of the instability of the synchronization manifold (measured by λL ) and from the other side there is that of the diffusion (measured by c). When the diffusion strength overcomes that of the instability, the systems synchronize.


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2.5. The stability of the synchronized motion Since the beginning of the studies on synchronization of chaotic systems, the issue of assessing the stability (or robustness) of the synchronous motion was considered the crucial point to be discussed, in order to furnish the proper conditions for a laboratory verification of the theoretical findings. This problem was therefore coped with soon after the first evidences of synchronized dynamics were published. In its turn, the problem of stability can be tackled in different ways, and different criteria can be established, depending of which specific condition one is interested to study. One of the most popular and widely used criterion is the use of the conditional Lyapunov exponents, that have been largely discussed already in the previous sections, and that constitute average measurements of expansion or shrinkage of small displacements along the synchronized trajectory. Once again, it has to be remarked that this analysis differs substantially from the classical Lyapunov analysis. Here, indeed, the name conditional refers to the fact that these exponents are explicitly calculated in the proximity of the synchronized manifold (i.e., under the condition that the system’s trajectory be close to the CS manifold). In practice, this means that the nonlinear trajectory, along which the variational Lyapunov equations are evaluated, is explicitly taken as to be inside the synchronization manifold. As so, this analysis is local in the sense that it takes into account how a perturbation evolves if applied close to the synchronous motion, but it cannot account for global stability properties, nor for global convergence to the synchronization manifold in the phase space. It is evident that the negativity of these conditional exponents represents a necessary condition for the local stability of the synchronized motion. If these exponents, indeed, are positive, this means that one will never observe the system in its synchronous motion, because perturbations in the proximity of the manifold would grow exponentially, and would have the effect to destroy synchronization. The problem can be formulated in its very general form, by addressing the question of the stability of the synchronization manifold x ≡ y, or equivalently by studying the temporal evolution of the synchronization error e ≡ y − x. Following equation (2.4), the evolution of e is given by e˙ = f x, s(t) − f y, s(t) , (2.7) where x and y represent the state vectors of the response system and its replica. Notice that equation (2.7) can be written in both PC and APD configurations, since it explicitly includes the driving signal s(t). The necessary condition for the CS state to be linearly stable (and therefore observable in the presence of an arbitrary small noise) is that the synchronization

18


manifold is asymptotically stable for all possible trajectories s(t) of the driving system within the chaotic attractor. This property can be proved by using stability analysis of the linearized system for small e e˙ = D x s(t) e (2.8) where Dx is the Jacobian of the vector field f conditioned to be evaluated onto the driving trajectory s(t). Normally, when the driving trajectory s(t) is constant (fixed point) or periodic (limit cycle), the study of the stability problem can be made by means of evaluating the eigenvalues of Dx or the Floquet multipliers [35]. However, in the general case that the response system is driven by a chaotic signal, only numerical evaluation of the corresponding Lyapunov spectrum can be made. And, indeed, various studies have concerned the evaluation of the Lyapunov exponents of system equation (2.8). In the context of driver–response coupling schemes, these exponents were called conditional Lyapunov exponents [18,23], or, alternatively, transversal Lyapunov exponents because they measure shrinkage or grown processes of the error signal, and, as so, they correspond to directions which are transverse (perpendicular) to the synchronization manifold x ≡ y [25, 36]. While the negativity of these conditional Lyapunov exponents provides a necessary condition for the stability of the synchronized state, it is far from being sufficient for warranting a synchronized motion to be set in the coupled systems. Indeed, relevant cases exist where these exponents are negative and yet the perfectly synchronized state is not observable, thus indicating that additional conditions should be fulfilled to warrant synchronization in a necessary and sufficient way [37]. In most cases, furthermore, the calculation of the conditional Lyapunov exponents cannot be made in analytic way, and therefore numerical algorithms should be used [38–40]. Another observation is in turn: the conditional Lyapunov exponents are obtained from temporal averages, and therefore they characterize the average stability over the whole manifold, but they do not guarantee that, due to possible variations of the local values of the exponents, some local instabilities take place in specific positions of the synchronization manifold. A second approach to the problem of stability of the synchronous motion can be followed by the use of the Lyapunov functions [23,30,41]. This method provides necessary and sufficient conditions for global stability. Unfortunately, the existence of such functions, as well as the methods to explicitly construct them, is an issue that is assessed only for a rather limited number of cases, while a general procedure (generically applicable to any dynamical system) to obtain these functions is not yet available.


19

The point of having local desynchronization events taking place within of the CS manifold was discussed in Ref. [36]. There, the synchronized behavior of two chaotic circuits coupled in a drive-response configuration is studied. Ref. [36] shows that long CS intervals may well be interrupted by brief, but persistent, desynchronization events. To demonstrate that the Lyapunov exponents do not prevent from local desynchronization events, the average distance from the CS manifold |X⊥ |rms and its maximum observed value |X⊥ |max can be measured. |X⊥ |rms is sensitive to global stability, while |X⊥ |max is sensitive to local stability of the CS state. In Figure 2.2, these quantities are reported together with the largest conditional Lyapunov exponent λ1⊥ against the strength of the coupling.

Figure 2.2. From Ref. [15]. The panel (a) reports the average distance from the complete synchronization manifold |X⊥ |rms and its maximum observed value |X⊥ |max as functions of the coupling strength in the experiment of Ref. [36] with unidirectionally coupled chaotic circuits, in which long epochs of complete synchronization were observed to be interrupted by persistent desynchronization events. The panel (b) reports the corresponding theoretical predictions for the stability of the synchronous state, by showing how largest conditional Lyapunov exponent λ1⊥ and the maximum transverse Lyapunov exponent of the most unstable invariant set η⊥ depend on the coupling strength.


20

In order to predict this intermittent loss of synchronization, the authors of Ref. [36] propose two different parameters whose negativity would determine the local stability of the synchronization manifold. One of them is the maximum transverse Lyapunov exponent of the most unstable invariant set η⊥ , whose dependence with c is also shown in Figure 2.2. Although this criterion rigorously and clearly predicts the synchronized state, its application may be difficult in practice, due to the infinite number of invariant sets where stability should be determined. The appearance of these local desynchronization event was called attractor bubbling and it has been characterized as a new type of intermittency associated to unstable invariant sets embedded within the synchronization manifold [42]. It is important to notice that such a kind of intermittency can be triggered by low levels of noise or slight parameter mismatches between the coupled systems.

2.6. The master stability function A very important extension of the concept of transverse stability to the case of multiple coupled systems is the so-called Master Stability Function. This section aims at presenting this formalism in its most general formulation, as the main results will be later extensively used also for the description of synchronization phenomena in networks of arbitrarily coupled identical systems. The Master Stability Function (MSF) approach was indeed originally introduced for arrays of coupled oscillators [43], and later extended to the case of a complex networks of dynamical systems coupled with arbitrary topologies [44–47]. 2.6.1. The case of time continuous systems We start by considering N coupled dynamical units, each one of them giving rise to the evolution of a m-dimensional vector field xi ruled by a set of ordinary differential equations x˙ i = Fi (xi ). The equation of motion reads: x˙ i = Fi (xi ) − σ

N

Gij H[xj ],

i = 1, . . . , N.

(2.9)

j =1

Here x˙ i = Fi (xi ) governs the local dynamics of the vector field xi ∈ Rm representing the state of the ith system, the output function H[x] is a vectorial function, σ is the coupling strength, and G is a coupling matrix, whose element i = j is different than zero only when the systems i and j are coupled. Notice that, depending on the specific choice of G, one here encompasses a very general class of coupled configurations, ranging from regular lattices of cou-

2.6. The master stability function

21

pled systems, to unidirectional chains, to networks of systems coupled through complex wiring’s topologies. In order to proceed with the analytic treatment, we will make the following explicit assumptions: • The systems are identical (Fi (xi ) = F(xi ), ∀i), identically coupled, and autonomous. This is tantamount to assume that the local dynamics F and the output function H do not explicitly depend on the index i, nor on time. • The coupling matrix G is a real symmetric zero-row sum matrix, i.e., the diagonal elements are given by Gii = − j =i Gij . The matrix G is therefore diagonalizable, and there exists a set of eigenvalues λi (of associated orthonormal eigenvectors [vi ]), such that Gvi = λi vi and vTj · vi = δij . Further, because the zero row condition, the spectrum of eigenvalues is entirely semi-positive, i.e., λi 0 ∀i. Though this assumption is not strictly necessary (we will see in the following that the validity of the approach have been extended to the case of nonsymmetric, or even nondiagonalizable, matrices), it will serve us for the sake of illustration of the main ideas that are behind the rigorous treatment. Taken together, these assumptions ensure that the coupling term in equation (2.9) vanishes when xi (t) = xs (t), ∀i, making the synchronization state an invariant manifold. Such a synchronized state xi (t) = xs (t), ∀i, with x˙ s = F(xs ) is a—possibly unstable—solution of equation (2.9). A major step forward is the observation that the linear stability analysis of equation (2.9) can be carried out by separating the topological and dynamical features [43]. As we will see momentarily, indeed, the assessment of the stability of the CS manifold involves the calculation of the spectrum of the coupling matrix G, and the derivation of the so-called Master Stability Function for the specific choice of the functions F and H. A necessary condition for stability the synchronization manifold is that the set of (N − 1) ∗ m Lyapunov exponents that corresponds to phase space directions transverse to the m-dimensional hyperplane x1 = x2 = · · · = xN = xs be entirely made of negative values. In order to calculate this set of conditional Lyapunov exponents, let δxi (t) = xi (t) − xs (t) = δxi,1 (t), . . . , δxi,m (t) be the deviation of the ith vector state from the synchronization manifold, and consider the m × N column vectors X = (x1 , x2 , . . . , xN )T and δX = (δx1 , . . . , δxN )T .


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The equation of motion for the perturbation δX can be straightforwardly obtained by expanding equation (2.9) in Taylor series of 1st order around the synchronized state, which gives

˙ = IN ⊗ JF(xs ) − σ G ⊗ JH(xs ) δX, δX (2.10) where ⊗ stands for the direct product between matrices, and J denotes the Jacobian operator. Solving this system can be rather complicated because of its (possible) high dimensionality. One can notice, however, that the arbitrary state δX can be written as δX =

N

vi ⊗ ζi (t)

i=1

[ζi (t) = (ζ1,i , . . . , ζm,i )], because the eigenvectors vi furnishes here a basis in RN . By operating the scalar product with vTj of the left side of each term in equation (2.10) (and because the matrix G is symmetric and therefore the eigenvector basis is orthonormal), one finally obtains a set of N blocks (the variational equations in an eigenmode form) for the coefficients ζi (t) that read ζ˙j = Kj ζj ,

(2.11)

where j = 1, . . . , N and Kj = [JF(xs ) − σ λj JH(xs )] is the evolution kernel. It is important to notice that each block in equation (2.11) corresponds to a set of m conditional Lyapunov exponents (the kernels Kj are calculated on the synchronization manifold) along the eigenmode corresponding to the specific eigenvalue λj . Because of the zero-row condition of the coupling matrix, one has that λ1 ≡ 0 and the associated eigenvector ±1 v1 = √ {1, 1, . . . , 1}T N entirely defines the synchronization manifold. All the other eigenvalues λi (i = 2, . . . , N ) and associated eigenvectors vi span instead all the other directions of the (m × (N − 1))-dimensional phase space transverse to the synchronization manifold. The necessary condition for stability of the synchronized state is therefore that all Lyapunov exponents associated with λi for each i = 2, . . . , N (the directions transverse to the synchronization manifold) are negative. As the eigenmode associated to the eigenvalue λ1 = 0 lies entirely within the synchronization manifold, the corresponding m conditional Lyapunov exponents equal those of the single uncoupled system x˙ = F(x). The synchronized state itself can therefore well have a positive Lyapunov exponent and be chaotic.


23

It is important to highlight once again that, for the sake of exemplification, we have up to now limited the discussion to the case of symmetric matrices. In fact, the very same criterion of all Lyapunov exponents associated with λi for each i = 2, . . . , N in equation (2.11) being negative constitutes a necessary condition for the stability of the synchronization manifold also for nondiagonalizable (yet zero-row sum) coupling matrices, as it has been recently shown in [48]. For an arbitrary (not necessarily symmetric) real coupling matrix, the eigenvalues spectrum is either real or made of pairs of complex conjugates, and the terms σ λi may take, in general, complex values [49]. We can therefore consider the following m-dimensional parametric equation ζ˙ = JF(xs )ζ − (α + iβ)JF(xs )ζ,

(2.12)

that, once coupled with the m-dimensional local nonlinear evolution of the networked system (˙xs = F(xs )), gives a set of m Lyapunov exponents for any choice of α and β. The surface (Λ(α, β)) in the parameter plane reporting the maximum of such exponents is called Master Stability Function [43], and fully defines the stability properties of the synchronization manifold. Given a coupling matrix, the stability of the different eigenmodes can be evaluated by the sign of Λ(α, β). Varying the real coupling terms in α + iβ can be interpreted as having an effect of damping the transverse perturbations of the synchronization manifold, whereas changes in imaginary terms have a rotation effect between different modes [50]. If all the eigenmodes (associated with the different λi , i 2) are stable (Λ < 0) the synchronous state is then linearly stable. To make an illustrative example, let us consider the case of a set of coupled chaotic Rössler oscillators [34]. The dynamics is ruled by equation (2.9), with x = (x, y, z), and

F(x) = F(x) = −y − z, x + 0.165y, 0.2 + z(x − 10) . The Master Stability Function is depicted in Figure 2.3 in the complex plane (α, β) for two type of couplings: H[x] = y and H[x] = x. For both the x and y-couplings Λ(0, 0) > 0 because this is just the case of uncoupled chaotic systems. For the y-coupling (Figure 2.3(a)), we observe that with increasing α the Master Stability Function Λ(α, β) drops below zero and remains negative even for large values of coupling. Therefore, a sufficiently large coupling would always guarantee in this case the stability of the synchronized state. For the x-coupling (Figure 2.3(b)), instead, Λ(α, β) is only negative within a finite region and stability is lost for large couplings strengths. In both examples

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Figure 2.3. The Master Stability Function (MSF) Λ = λmax in the parameter space (α, β) for the Rössler oscillator [34] coupled through a y-coupling (a) and x-coupling (b) schemes (H[x] = y and H[x] = x, respectively). Notice that the regions of stability Λ(α, β) < 0 are given by the surfaces under the dark plane at Λ = 0. While in all cases Λ(0, 0) > 0 (the uncoupled system is generating a chaotic dynamics), it is worth noticing that in panel (a) Λ(α, β) drops below zero with increasing α, and remains negative even for large values of coupling. At variance, in panel (b), Λ(α, β) is only negative within a finite region of parameter space.

of Figure 2.3, the regions of stability Λ(α, β) < 0 are given by the surfaces under the plane at Λ = 0. Generically, therefore, there exists a bounded region of stability in the complex plane (symmetric in the imaginary values about the real axis) where Λ(α, β) < 0 and an arbitrary increasing of the coupling strength may induce a destabilization of the synchronous state [43,44,50].


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2.6.2. The master stability function for coupled maps All the results discussed in the previous section can be straightforwardly extended to arbitrary ensembles of coupled time-discrete systems (coupled maps) [50]. Let us consider a network of N coupled maps, which dynamical evolution is ruled by xit+1

N

i j = f xt − σ Gij H xt ,

i = 1, . . . , N,

(2.13)

j =1 j

where xt+1 = f(xit ) governs the local dynamics of each map, and the output function H[x] defines the coupling function. As for the time continuous case, σ and G define the coupling strength and the coupling matrix (with eigenvalues 0 = λ1 λ2 · · · λN ), respectively. Let us consider xit = xst , ∀i, to be the synchronized state of coupled variables. The linear stability of the synchronization manifold is governed by the variational equations of equation (2.13) which have the following block diagonalized form

i ηt+1 (2.14) = exp Λ(σ λi ) ηti = Jf xst − σ λi JH xsn ηti , where Λ(λi ) is the Lyapunov exponent. For illustration purposes (the stability of coupled map networks are studied in detail in Refs. [45,50–57]), let us consider the coupling function H(x) = f(x). Under this assumption, JH = Jf and the system (2.14) reduces to i ηt+1 (2.15) = Jf xst − σ λi Jf xst ηti . It turns out that the explicit conditions of stability for chaotic maps can be obtained as: 1 − exp(−μmax ) 1 + exp(−μmax ) 0, three possible behaviors of Λ(ν) can be produced in the vicinity of the origin, defining three possible classes for the choice of the local function F(x) and of the coupling function H(x): (I) Λ(ν) is a monotonically increasing function, (II) Λ(ν) is a monotonically decreasing function that intercepts the abscissa at some νc 0, and (III) Λ(ν) is a V-shaped function admitting negative values in some range 0 ν1 < ν2 . The three classes of Master Stability Function are sketched in Figure 2.4. It is easy to understand that both cases (I) and (II) of Figure 2.4 correspond to rather trivial situations. Indeed, case (I) is tantamount to say that one never stabilizes synchronization in the system for that choice of F(x) and H(x) (for all σ

Figure 2.4. The three possible behaviors of Λ(ν) that can be produced in the vicinity of the origin, and that define the three possible classes for the choice of the local function F(x) and of the coupling function H(x): (I) Λ(ν) is a monotonically increasing function, (II) Λ(ν) is a monotonically decreasing function that intercepts the abscissa at some νc 0, and (III) Λ(ν) is a V-shaped function admitting negative values in some range 0 ν1 < ν2 .


27

values and for all possible eigenvalues’ distributions, the product σ λi always leads to a positive maximum Lyapunov exponent, and therefore the synchronization manifold is always transversally unstable). The very opposite situation arises for functions F(x) and H(x) giving Master Stability curves as the one of the case II in Figure 2.4. There, the system admits always synchronization for a large enough coupling strength, regardless on the topology of the coupling configuration (given any eigenvalue distributions it is indeed sufficient to select σ > νc /λ2 (where νc is the intersection point of the Master Stability Function with the ν axis) to warrant that all transverse directions to have associated negative Lyapunov exponents). In this latter case, once fixed x˙ = F(x) and H(x) (which fix the value of νc ) the effect of the connection topology is only to rescale (by means of λ2 ) the threshold for the appearance of a synchronous state. A nontrivial and interesting situation is case (III), which by the way corresponds to a large class of possible choices of F(x) and H(x) [44]. Here, Λ(ν) is negative in a finite parameter interval (ν1 , ν2 ) (with ν1 = 0 when F(x) supports a periodic motion). The stability condition is then satisfied for some σ when λN /λ2 < ν2 /ν1 . A measure of synchronizability is then the ratio λN /λ2 between the largest and the second smallest eigenvalue in the spectrum of the coupling matrix: the more packed the eigenvalues of G are, the higher is the chance of having all Lyapunov exponents into the stability range for some σ [44]. In other words, if the ratio λN /λ2 (that, by definition, is larger than unity) is small enough, the eigenvalue spectrum of the coupling matrix comes out to be all packed around the same value, and therefore there is a higher chance that, when multiplied by the same σ , it will enter the stability region. If, on the contrary, the coupling configuration is such that λN /λ2 is very large, it may happen that for the same σ for which σ λ2 > ν1 one has that σ λN > ν2 , and therefore the system will always present at least one direction of instability for the synchronization manifold. In this latter case, one will be unable to match the necessary condition for stability, and therefore the specific coupling configuration would be unable to produce a synchronous motion. It is important to notice that, in principle, one can also have a fourth case, where Λ(ν) is an oscillating function that intersects multiple times the ν axis, this way featuring multiple stability regions. This latter situation, however, is not conceptually different from the class III Master Stability Function case, as far as one is interested in studying synchronization events occurring at low values of the coupling strength (i.e., close to the origin of Figure 2.4). The only difference will be that the system here will experience a series of consecutive de-synchronization and re-synchronization processes, as one further increases the coupling strength σ .


28

The situation is a bit more complicated when the coupling matrix G is asymmetric and diagonalizable. In such a case, the spectrum of G is contained in the complex plane (λ1 = 0; λl = λrl + iλil , l = 2, . . . , N ), and one has to study the parametric equation (2.12) in the full complex plane (α, β). Nevertheless, an ordering of G’s eigenvalues can be done for increasing real parts. Gerschgorin’s circle theorem [58,59] asserts that G’s spectrum in the complex plane is fully contained within the union of circles (Γi ) having as centers the diagonal elements of G (di ), and as radii the sums of the absolute values of the other elements in the corresponding rows:

{λl } ⊂ Γi di , |Gij | . i

j =i

Notice that if the coupling matrix is made of real entries, in all cases the circles have centers lying on the real axis. If one further supposes the diagonal elements of G to be conveniently normalized to 1 in all possible cases, and that all nonzero off diagonal entries of G are negative, then relevant mathematical and physical consequences arise. Physically, this normalization process prevents the coupling term from being arbitrarily large (or arbitrarily small) for all possible coupling topologies and system’s sizes, thus making it a meaningful realization of what happens in several real world situations (such as neuronal networks) where the local influence of the environment on the dynamics does not scale with the number of connections. Mathematically, since G is a zero row-sum matrix (and di = j =i |Gij | follows from the extra assumption that all nonzero off diagonal elements are negative), this warrants in all cases and for all system’s sizes that G’s spectrum is fully contained within the unit circle centered at 1 on the real axis (|λl − 1| 1, ∀l), giving the following inequalities: • 0 < λr2 · · · λrN 2, and • |λil | 1, ∀l. This latter property is essential to provide a consistent and unique mathematical framework within which one can formally assess the relative merit of one coupling topology against another for optimizing the propensity for synchronization of a system, regardless on the specific properties of the local dynamics. By calling R the bounded region in the complex plane where the master stability function Λ((α, β)) provides a negative Lyapunov exponent, the stability condition for the synchronous state is that the set {σ λl , l = 2, . . . , N } be entirely contained in R for a given σ . This is best accomplished for connection topologies that simultaneously make the ratio λrN λr2


and

29

M ≡ max λil l2

as small as possible. We remark that also in Refs. [60–62], the value of the coupling strength necessary to synchronize a network of arbitrarily coupled identical systems was proposed to be ruled by the structure of the coupling matrix, and by the coupling strength necessary to synchronize two elements of the network. In particular, the so-called Wu–Chua conjecture implies that the coupling threshold for synchronization in such an array is 2σ2 , |λ2 | where σ2 is the synchronization threshold for synchronization of two oscillators. We stress that this conjecture is only valid under the assumption that the stability of the transversal mode associated with λ2 induces the stability of all other modes. This implies that a network of coupled identical oscillators can always be synchronized for sufficiently high coupling strengths. While the analysis of networks formed by coupled Lorenz or Chua’s oscillators has confirmed this conjecture, other relevant cases (see the examples in Figure 2.3) exist where an increasing of the coupling strength results in a destabilization of the synchronous state. Master stability function arguments are currently used as a challenging framework for the study of synchronized behaviors in complex networks, especially for understanding the interplay between complexity in the overall topology and local dynamical properties of the coupled units. At the same time, more sophisticated approaches have been developed to assess conditions for synchronization in a network of coupled units. An example is the recent connection graph stability method [63], that combines Lyapunov function approaches with graph theory, leading to the determination of a rigorous bound for the minimum coupling strength needed for yielding global synchronization. This approach allows also for describing some cases, where the coupling matrix G is time-dependent, i.e., G = G(t). Furthermore, recent studies have approached the relationships between symmetry group theory and stability of the synchronized (or patterned) solutions in networks with complex topologies [64], revealing that the symmetry groups in the architecture of complex networks can determine constraints for the appearance or stability of a given network solution. Many relevant situations, as, e.g., pulse-coupled networks of bursting neurons, correspond to inherently nonlinearly coupled systems. Therefore, the attention has also started to concentrate on nonlinearly coupled units, with the same aim of relating the topology of the network to the synchronization properties. σc =

30


For instance, when such kinds of networks are very homogeneous in the degree distribution (when all nodes have the same degree k), a recent study [65] has shown that the onset and stability of the synchronous state only depends on the number of signals each neuron is receiving, regardless on the details of the network topology. Even though such a statement strongly depends on the assumption that all units have the same number of connections (and therefore it does not apply to general networks), the fact that synchronization in ensembles of neurons (each one of them receiving k inputs) can be ensured by means of a single condition is a relevant result. We will return to the Master Stability Function approach in the chapter describing synchronization in complex networks. In the next chapter, instead, we pass to discuss the case of coupled nonidentical systems, for which synchronization states other than complete synchronization may be defined.

Chapter 3

Nonidentical Systems

He recorrido hasta el último amanecer para hallar el rumbo de tus deseos y sin embargo muchas fueron las incertidumbres de mis caminos, y muchos los senderos desviadores que mis erratiles pasos se esforzaron en cumplir. Pero las flores no eligen sus colores ni los ríos los álveos que escavan, y nadie es dueño de su destino. Así, todo lo que creía y que no he sido atraviesa repentinamente mi pensamiento, calmando la quemazón del sediento y regalando la dulzura del olvido. Hasta que al final seremos lo que en el alborear de mundo fuimos.1 Immediately after the first studies on synchronization of chaotic identical systems, it became evident the need of extending the interest to the case of nonidentical systems, as experimental and real systems are, in fact, never identical. An important preliminary observation is that, as soon as diversity between the coupled systems is considered, in most circumstances, the complete synchronization manifold x = y ceases to exist as an invariant manifold. Therefore, different concepts and types of synchronization behavior in coupled nonidentical low-dimensional chaotic systems have to be defined. 1 I’ve run after up to the last morning, to find out the direction of your wishes, and many have been

the dubitations of my roads, and many the wrong paths that my week foot tried to make. But the flowers do not choose their colors, nor the river the path they’re eroding, and no one is owner of his destiny. Then, all I believed and has been not crosses rapidly my mind, cooling the heat of the thirsted, and giving the mildness of the oblivion. Up to when we will be what at the beginning of the world we were. 31

32


In particular, we will concentrate on phase synchronization [6], a rather weak degree of synchronization characterizing the behavior of coupled chaotic oscillators. This synchronization regime consists in the fact that, starting from uncoupled nonsynchronized oscillatory systems, an increase in the coupling strength yields a collective dynamics where suitably defined phases of the chaotic oscillators become locked, without implying a substantial correlation in the corresponding amplitudes. A central concept that will accompany all the discussion on phase synchronization is that of phase coherence of a chaotic oscillator. Indeed, a perfect phase synchronization between two coupled oscillators could only occur when the chaotic oscillators are phase coherent, i.e., when the distribution of return times to the Poincaré section is well peaked around a mean value, or, alternatively, when the power spectrum of the chaotic system is well peaked around a dominating frequency. In the opposite case, i.e., when the chaotic oscillations cover a very broad range of time scales, the definition of instantaneous phase becomes problematic, and phase synchronization epochs are in general interrupted by intermittent phase slips, giving rise to a phenomenon termed imperfect phase synchronization, where the phase locking is not perfect, i.e., not all frequencies of one chaotic oscillator are locked to those of the other oscillator. In general, phase synchronization describes the weakest for of synchronization, and it emerges for already very low values of the coupling strength. At larger values of coupling strengths, nonidentical systems may display other regimes of synchronization, that are characterized also by strong correlations in the chaotic amplitudes. For instance, this is the case of lag synchronization [8], where the states of the two chaotic oscillators become effectively identical in time, when shifted by a proper lag time. On its turn, the lag time progressively vanishes, as the coupling strength increases, up to when eventually its value becomes zero, and the pair of coupled systems attain a regime of almost complete synchronization, where the temporal behavior of the difference between the states of the two systems displays very small oscillations around a zero value. Finally, when fairly different chaotic systems x˙ = f(x) and y˙ = g(y) are strongly enough coupled, the collective dynamics may collapse into a subspace of the whole phase space (x, y). This new manifold is not in general the complete synchronization manifold x = y, but it is described by a more complicated time independent functional relationship, e.g., y = h(x). The appearance of such a synchronized motion takes the name of generalized synchronization [9,23]. Indeed, the manifold y = h(x) can be regarded as a generalization of the complete synchronization manifold, that would correspond to a function h equal to the identity function. All these processes, and the relevant features characterizing the transitional stages to them, will be described in details in the following paragraphs. This chap-

3.1. Phase synchronization of chaotic systems

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ter ends with a large review of the main methods and tools that are commonly used to detect, identify and characterize synchronous behavior from the acquisition of multivariate data.

3.1. Phase synchronization of chaotic systems 3.1.1. Synchronization of periodic oscillators We start by discussing the details of phase synchronization between coupled oscillators. In order to do so, we begin with briefly summarizing the classical notion of phase locking of coupled periodic oscillators. When two periodic oscillators are coupled, n : m phase synchronization is defined as the hooking of the instantaneous phases φ1,2 (t) to the corresponding ratio n : m (being n and m integers). The mathematical condition associated with this definition is that the time evolution of the absolute value of the difference between the two instantaneous phases (multiplied respectively for n and m) remains bounded in time, i.e., nφ1 (t) − mφ2 (t) < const. A direct consequence of such phase locking is that the corresponding average frequencies 1 = lim T →∞ T

T

φ˙ i (τ ) dτ

(i = 1, 2)

0

are also locked, i.e., n − m = 0, and therefore synchronization of the phases implies here perfect synchronization of the oscillators’ mean frequencies. It is relevant to observe that the above definitions do not imply specific conditions on the amplitudes of the oscillations. And, indeed, also when phase locking is set, the oscillators’ amplitudes are free to differ from one another. This is the main reason why such a phenomenon requires only a small coupling value, able to lock the oscillators’ phases, but not strong enough to affect the time evolution of the oscillators’ amplitudes. In order to give a more quantitative description of phase synchronization of weakly coupled periodic oscillators, one can refer to the dynamics of the phase difference θ = nφ1 (t) − mφ2 (t), which generally obeys θ˙ = ω − C sin θ,

(3.1)

being ω = nω10 − mω20 the difference between the natural frequencies of the oscillators, and C the coupling strength. In this context, a perfect phase locking is


34

obtained when the parameters satisfy ω C 1.

(3.2)

Equation (3.2) defines in its turn a specific region in the parameter space where phase locking is attained, and which is known as the Arnold tongue [66]. The region delimits the parameter area for which the fixed point ω C is stable. The fixed point θ0 corresponds to a minimum of the washboard potential θ0 = arcsin

V (θ ) = −θ ω − C cos θ. When these properties are described for experimental or natural systems, it is necessary, however, to further discuss the effects of noise, whose presence is unavoidable in real systems. In general, the presence of a noisy source ξ(t) affecting the evolution of the systems destroys perfect phase synchronization and frequency locking. Indeed, with the addition of noise, the dynamics of the phase difference is described by θ˙ = ω − C sin θ + ξ(t).

(3.3)

The usual approach to provide a detailed analytical description of the noisy case consists in assuming the noise term ξ(t) to be Gaussian delta-correlated [67] and studying the associated Fokker–Planck equation [68]. As one would expect, the noise term causes a perturbation of the phase locked state. As a consequence, the phase difference now fluctuates around the minimum of the washboard potential V (θ ), and can occasionally climb over the energy barrier passing from a local minimum to the neighboring one, as illustrated in Figure 3.1(a). The result of this jumping process is that one observes noise-induced 2π phase slips (Figure 3.1(b)), i.e., sudden 2π jumps in the temporal evolution of the phase difference. It has to be remarked, however, that not always noise has a destructive effect on coupled nonlinear systems. Indeed, when the noise acts on a nonlinear system, it can even produce cooperative and constructive effects, as, e.g., it can enhance synchronization and can even induce synchronized states in uncoupled systems. These phenomena will be described later in Chapter 4, in the context of coupled chaotic systems. In order to illustrate the above scenario, we here consider the coupling between a nonlinear Rössler oscillator and a driving periodic signal. The system is described by:


35

Figure 3.1. From Ref. [15]. Left panel: systematic plot of the washboard potential V (θ) = −θω − C cos θ for the system (3.1). Right panel: effect of a noise term causing a perturbation of the phase locked state. The phase difference now fluctuates around the minimum of the washboard potential V (θ), and can occasionally climb over the energy barrier passing from a local minimum to the neighboring one (see the left panel). The resulting effects are noise-induced 2π phase slips, i.e., sudden 2π jumps in the temporal evolution of θ.

x˙ = −ωy − z + E sin(Ωe t), y˙ = ωx + ay, z˙ = f + z(x − c).

(3.4)

When the parameters of equation (3.4) are set to the values ω = 0.97, f = 0.2, c = 10 and a = 0.04, the unforced (E = 0) Rössler oscillator exhibits a periodic oscillatory motion at a frequency Ω = 0.981. Figure 3.2(a) shows that the frequency of this periodic motion can be locked in the ratio n : m = 1:1 to that of the external periodic signal Ωe = 1.0, already

Figure 3.2. From Ref. [15]. Left panel: Synchronization of periodic oscillation (solid line) to a weak periodic driving signal (dotted line). Notice that the frequency of the periodic motion is here locked in the ratio n : m = 1:1 to that of the external periodic signal, already for a very weak amplitude. Right panel: the whole Arnold tongue in the parameter space (see text for definition) for 1:1 phase synchronization, delimiting the parameter region where perfect phase locking is produced.

36


for a very weak amplitude (E = 0.4). The whole Arnold tongue in the parameter space for 1:1 phase synchronization is, instead, shown in Figure 3.2(b), delimiting the whole parameter region where perfect phase locking is produced. 3.1.2. Phase of chaotic signals What discussed in the previous section about the classical notion of phase synchronization in the case of coupled periodic oscillators can be extended to the case of coupled chaotic oscillators [6]. It is easy to verify that equation (3.4) for different values of the parameter a (a = 0.165) gives rise to autonomous chaotic oscillations, as it can be seen in Figure 3.3(a), that reports a snapshot of the time series x(t) in the case of E = 0. The first, and most important, problem emerging in this context is that of suitably defining and uniquely determining the time-dependent amplitude A(t) and instantaneous phase φ(t) for a nonperiodic signal. For the sake of clarity, it must be said at this stage that a unique definition for the instantaneous phase of a chaotic system has not yet been provided, and in most cases the approach one has to follow crucially depends on the specific properties of the chaotic system itself. While this constitutes undoubtedly a limitation to the rigorous approach of phase synchronization in nonperiodic oscillators, some operative strategies can be adopted to properly describe synchronization phenomena for relevant classes of chaotic oscillators. In the following we will discuss and compare three possible strategies to operatively assign a quantity that corresponds to an instantaneous phase in the case of chaotic oscillators. As we will see, none of this strategy can be assumed as a rigorous definition for the instantaneous phase. Rather, each one of them has

Figure 3.3. From Ref. [15]. Left panel: the temporal chaotic evolution of the chaotic Rössler oscillator. Right panel: the phase of the chaotic signal, as measured with equation (3.5) (solid line), equation (3.7) (dashed line) and equation (3.8) (dotted line). Notice that, for this phase coherent case, the three measurements give essentially the same quantity.


37

the meaning of providing an operative measure that, under certain circumstances, yields a quantity that can be associated to the phase of the chaotic oscillator. The first strategy to calculate phases of chaotic oscillators is the analytic signal approach introduced by Gabor [69]. Given an observed scalar time series s(t), the analytic signal ψ(t) is a complex function defined by ψ(t) = s(t) + j s˜ (t) = A(t)ej φ(t) ,

(3.5)

where the function s˜ (t) is the Hilbert transform of s(t) 1 s˜ (t) = P.V. π

∞

−∞

s(τ ) dτ, t −τ

(3.6)

and P.V. stands for the Cauchy principal value for the integral. This method sorts a quantity that can be reasonably associated to an instantaneous phase in all cases for which the power spectrum of the oscillator is well peaked around a given frequency (i.e., for phase coherent oscillators). Given that the Hilbert transform implies a convolution integral, its output cannot be, however, meaningfully associated to an instantaneous phase in all cases in which, e.g., the spectrum is wide, or the system shows multiple incommensurable frequency peaks in its spectrum. A second, geometrical, strategy can be adopted in all cases in which the trajectory of the chaotic oscillator has a proper rotation around a certain reference point in phase space. In this latter case, a measurement of instantaneous phase can be provided in an intuitive and straightforward way. For example, in the Rössler chaotic oscillators at a = 0.165, the projection of the chaotic attractor in the x–y plane looks like a smeared limit cycle (see Figure 3.4(a)), and therefore the instantaneous phase can be simply defined as the angle φ(t) = arctan y(t)/x(t) . (3.7) In other circumstances, one could operate a proper projection of the flow into a proper plane containing a unique rotation center, and use the rotation angle as a direct measure of instantaneous phase. The reader should be warned, however, that this technique requires necessarily the existence of a unique rotation center. If multiple rotation centers exist, the angle with respect to any one of them cannot be taken as a proper measure of the instantaneous phase, as this measure does not satisfy the most important property for a phase measure, that consists in increasing 2π at each oscillation. Furthermore, individuating the proper phase space projection under which a unique rotation center is found may generically comes out to be a very nontrivial task, and therefore the application of such a strategy can be limited by the concrete difficulty of finding the proper projection in phase space. A third strategy to measure an instantaneous phase of a chaotic flow is based on an appropriate Poincaré section with which the chaotic orbit crosses once for


38

Figure 3.4. From Ref. [15]. Projection of the chaotic Rössler attractor on the x − y plane. The left panel illustrates the case of the phase coherent attractor. In this case, the instantaneous phases can be easily calculated with the analytic signal method, the Poincaré section (e.g., the heavy dashed line in the figure) or simply the rotation angle. The resulting phase dynamics is very coherent, i.e., the return time τk − τk−1 has only a rather narrow distribution. At variance, the right panel reports the funnel chaotic Rössler attractor, in which a suitable measurement of phase variables for the system is complicated by the fact that the attractor does not show a unique center of rotation.

each rotation (Figure 3.4(a)). Successive crossing with the Poincaré section can be associated with a phase shift of 2π, and the basic idea is that all phases in between are computed with a linear approximation. The phase definition in this case is then t − τk−1 , τk−1 < t τk , φ(t) = 2πk + (3.8) τk − τk−1 where τk and τk−1 are the times for the kth and (k −1)st crossings of the trajectory with the Poincaré section, respectively. In particular, one can choose as the Poincaré section that corresponding to the successive maxima or minima of the chaotic time series. This is because, given a generic one-dimensional oscillatory signal x(t), the associated phase space can be fully reconstructed by using the coordinates dx(t) d 2 x(t) , ... , , (3.9) dt dt 2 with as many derivatives as needed to exhaust the dimensionality of the system. In such a reconstructed phase space one can then define the Poincaré section by noting the points obtained when the orbit crosses the surface dx(t)/dt = 0. This means that phase can be defined equivalently by examining the maxima or minima of the scalar chaotic time series without having to reconstruct the dynamics in a higher dimensional phase space. This method, however, is not always effective. For instance, in the presence of noise, the correct determination of minima and maxima of a signal can be strongly hampered, and often the application x(t),


39

of noise filtering may give a spurious measurement of the maxima or minima location, or, alternatively, it can even wash out existing maxima and minima. When one is dealing with a phase coherent chaotic oscillator, the three strategies gives essentially the same measurement of the instantaneous phase. For instance, we report in Figure 3.3(b) the instantaneous phases calculated in these different ways for the chaotic trajectory of Figure 3.3(a). For this specific case, it is then evident that the three definitions provide quantities that are in very good agreement to each other. As we will see later on, however, not all chaotic oscillators are phase coherent, and there are circumstances where one is forced to use one of the above definitions instead of another, depending on the specific properties of the chaotic signal, as the three definitions do not provide a measure of the same quantity. It is important to emphasize that this limitation will not affect essentially the property of phase synchronization of chaotic oscillators. In spite of the existence of different definitions, indeed, the instantaneous phase of a system is a monotonously increasing function of time. Both equation (3.5) and equation (3.7) show, however, that the increase in phases is not uniform, and that it is, instead, affected by pronounced fluctuations in the amplitude of the signal (see insert in Figure 3.3(b) for t ∼ 55). These fluctuations around the average linear increase can be accounted for dy defining a phase diffusion coefficient Dφ , given by 2 φ(t) − φ(t) = 2Dφ t, (3.10) where · denotes an ensemble average. The coefficient Dφ measures what is called the degree of phase coherence of the chaotic signal. In particular, if Dφ 1, the system is called phase coherent, i.e., its instantaneous phase is essentially growing in time in a linear way, as fluctuations around its average frequency are small. If, instead, Dφ is of the order or larger than unity, the system is phase incoherent. In this latter case, the fluctuations of the instantaneous phase around its linear growth are very pronounced. For example, the case of the system in Figure 3.3(a, b) displays fluctuations of the phase around the linear increase that are almost invisible. Therefore, in this case Dφ 1 and the system falls into the class of phase coherent systems. Another important point to discuss concerns the relationship between the instantaneous phase of an autonomous chaotic oscillator and the zero Lyapunov exponent appearing in the Lyapunov spectrum [6], which corresponds to the translation dx(t) along the chaotic trajectory. If one considers a system whose chaotic flow in the phase space displays a proper rotation around a given reference point, then dx(t) can be uniquely mapped to a shift of the phases dφ(t) of the oscillator. Due to such a connection, it appears evident that a synchronization of the instantaneous phases of two chaotic oscillators can be revealed by monitoring the Lyapunov spectrum, and looking at


40

the transition in parameter space wherein one zero Lyapunov exponent becomes negative. This, in its turn, corresponds to the fact that the two original oscillators (displaying initially independent phases and therefore giving rise to two independent zero Lyapunov exponents) set in a regime characterized by a unique phase (a unique zero Lyapunov exponent), wherein the phase of one system is slaved to the phase of the other. Most of the discussion we have made before concerns the case of phase coherent systems. When one deals with a system which is far from being phase coherent, i.e., it shows a wide distribution of frequencies in the spectrum, the individuation of a suitable measure for the phase variable of the chaotic oscillations is a problematic task, which has not been yet satisfactorily resolved. To illustrate the problems connected with such a task, let us refer to an example, the so-called funnel chaotic attractor of the Rössler oscillator (obtained for a = 0.25), that is shown in Figure 3.4(b). One immediately observes that the chaotic trajectory in this case does not cycle the unstable fixed point in all rotations. As a consequence, measuring the instantaneous phase by means of equation (3.7) would result in a quantity that would not increase monotonously with time. On the other hand, a proper Poincaré section for this specific case (a section that would cross once with the chaotic trajectory at each rotation) has not been yet rigorously identified, and therefore also the application of equation (3.8) is strongly limited. On the other hand, the funnel Rössier attractor shows a very wide spectrum of frequencies, and therefore also the analytic approach of equation (3.5) cannot be used. The definition of instantaneous phase can be encompassed within a more rigorous mathematical formalism, following what suggested in Refs. [66,70]. Let us, indeed, consider a chaotic oscillatory system x˙ = F(x). One can associate to it a phase φ(t) having the property of monotonically increasing in time with a natural period T such that φ(T + t) − φ(t) < η 1. (3.11) This statement is formally equivalent to consider a small phase diffusion Dφ . In particular, Ref. [70] has proved that, if φ(t) is strictly increasing with time, then one can define a set of suitable coordinates R and Φ [(R, Φ)], in the neighborhood of the chaotic attractor, so that ˙ = F(R, Φ), R Φ˙ = 1 + δ(R, Φ)

(3.12)

where Φ is T -periodic. One immediately realizes that, by means of the above transformation in coordinates, the phase dynamics becomes similar to that of a periodic orbit, except for the addition of the term δ(R, Φ) that is responsible for the coupling of the amplitude and phase coordinates, and that shows a sensitivity to the R variables. Phase


41

coherent systems, in this framework, correspond to the case where such a term is T small, e.g., 0 δ(R, φ) dφ = O(η) with η 1. In particular, Ref. [70] has fully developed an analytical approach for the quantitative description of the phase locked states, providing sufficient conditions for phase-locking to occur in a phase coherent system, and has applied the formalism to a chaotic electric circuit model which can be viewed as a piecewise linear simplification of the chaotic Rössler oscillator. 3.1.3. Phase entrainment of externally driven chaotic oscillators After having discussed how to assign a proper measure for the instantaneous phase of a chaotic system, we now move to show how phase synchronization of a chaotic oscillator with a periodic external driving signal can be detected, and we will exemplify our discussions with reference to the system equation (3.4) in the chaotic regime (a = 0.165). As one can see from Figure 3.5, when the instantaneous phase of the chaotic oscillator is synchronized with that of the driving signal, a stroboscopic plot of the system subspace (x, y) at each period of the driving signal results in a distribution of points that are restricted to a rather small arc area of the chaotic attractor. On the opposite, when there is not phase synchronization between the chaotic oscillator and the driving signal, the distribution of the points that represent the system subspace (x, y) (stroboscopically observed at each period of the driving signal) spans uniformly the whole chaotic attractor.

Figure 3.5. From Ref. [15]. Stroboscopic plot of the Rössler system state (x, y) (filled cycles) at each period of the driving signal (equation (3.4)) (the dotted background is the unforced chaotic attractor) for E = 0.15, Ωe = 1.0 (left panel, phase synchronization) and E = 0.15, Ωe = 1.02 (right panel, outside of the phase synchronization regime). Notice that, when the instantaneous phase of the chaotic oscillator is synchronized with that of the driving signal, the distribution of the stroboscopically observed points is restricted to a rather small arc area of the attractor. At variance, when the phases are not synchronized, the distribution of points spans uniformly the whole attractor.

42


Figure 3.6. From Ref. [15]. The synchronization region of the chaotic Rössler oscillator by an external periodic force (equation (3.4)). Notice that, for such a phase coherent system, the synchronization region in parameter space comes out to be very similar to the Arnold tongue for the periodic oscillators that was reported in Figure 3.2(b).

The synchronization region in parameter space shown in Figure 3.6 comes out to be very similar to the Arnold tongue for the periodic oscillators shown in Figure 3.2(b). These properties were firstly reported in Ref. [71] and studied more intensively in Ref. [67]. Again, the reason for the similarity between these two cases has to be found in the phase coherence nature of the chaotic oscillations, as we already discussed that the phases of phase-coherent chaotic oscillators increase almost linearly in time. In order to gather a more grounded and rigorous understanding of the similarity between a chaotic phase coherent oscillator and a periodic oscillator, one can follow the approach of revealing phase synchronization of chaotic systems by studying the phase locking features in terms of the set of unstable periodic orbits embedded within the chaotic attractor [72,73]. A chaotic trajectory, indeed, is known to visit at all times the neighborhood of a subjacent skeleton of unstable periodic orbits, jumping erratically in time from the vicinity of one orbit to the vicinity of another orbit. It is therefore, natural, to try to understand the phase synchronization phenomenon of a chaotic system by inspecting the phase locking properties of each one of such unstable periodic orbits. If one performs a stroboscopic recording of the amplitude x and phase φ of the chaotic signal at each period of the external forcing, one would get a twodimensional map, relating the values of x and φ recorded at the (n + 1)st period with those recorded at the nth period [72] x(n + 1) = f x(n), φ(n) ,

φ(n + 1) = φ(n) + ω + cos 2πφ(n) + g x(n) . (3.13) Notice that the second of equations (3.13) is essentially the circle map, which in this case is coupled to a chaotic map f . Furthermore, in this representation, ω


43

stays for the initial difference in the frequencies of the chaotic oscillator and of the driving force, while represents the coupling strength which is proportional to the amplitude of the driving force, and g(x(n)) corresponds to the nonuniformity of the phase rotations in the chaotic oscillators, as a result of the chaotic fluctuations of the amplitude x. The average growth rate of the phase

Ω = lim φ(n) − φ(0) /n n→∞

corresponds to a phase rotation number. The condition Ω = 0 individuates the synchronization of the chaotic oscillator to the external force. To exemplify our reasoning and without lack of generality, one can consider the chaotic tent map

f (x, φ) = 1 − 1.9|x| + 0.05 sin 2πφ(n) with g(x) = 0.05x [72]. In this framework, one can then represent the attractor in terms of the infinite set of unstable period orbits embedded in it, and one notices that the N th of such periodic orbits will be characterized by a period T ≈ T0 N , being T0 the average return time to the Poincaré section. Different periodic orbits however will show return times that fluctuate around multiples of T0 , due to the presence in the map system of the term g(x). Because of these fluctuations, each individual unstable periodic orbit will display a specific phase-locking region in parameter space under the periodic external forcing (i.e., it will give rise to its own Arnold tongue that will be different from the Arnold tongue of any other unstable periodic orbit), as it can be seen in Figure 3.7. In this representation, it is evident that the region in parameter space of phase synchronization of the chaotic oscillator will be given by the overlapping region (if any) of all the Arnold tongues of the periodic orbits. In this overlapping region, indeed, the chaotic trajectory will be visiting closely unstable periodic orbits that are “all” phase locked with the external driving signal. The existence of such an overlapping region of all the Arnold tongues of periodic orbits is a property of phase coherent chaotic oscillators. It has to be expected then, that when a chaotic oscillator is not phase coherent, there is not a region of parameter space in which all the Arnold tongues of the corresponding periodic orbits overlap. Still in these conditions, there might exist a region in which most of the Arnold tongues overlap, and the chaotic system will display an “imperfect” phase synchronization, giving rise to a phase locked motion as far as the chaotic trajectory visits closely one of the periodic orbits whose Arnold tongue stays in the overlapping region, but unavoidably will display 2π phase jumps all the times at which the chaotic trajectory visits the vicinity of one periodic orbits whose associated Arnold tongue is not inside the overlapping region.

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Figure 3.7. Phase-locking regions for periodic orbits with period 1-5 in the system of equation (3.13). Each individual unstable periodic orbit displays a specific phase-locking region in parameter space under the periodic external forcing. The region of phase synchronization of the chaotic system (colored in gray) corresponds to the overlapping region of all the Arnold tongues of the periodic orbits. In this region, indeed, the chaotic trajectory will be visiting closely unstable periodic orbits that are “all” phase locked with the external driving signal. Reprinted with permission from Ref. [12]. © 1999 The American Physical Society

This qualitative picture of phase synchronization has been quantitatively confirmed for the first time by the precise calculation of the Arnold tongues of the different unstable periodic orbits embedded in the continuous-time Rössler attractor [73]. In particular, it has been shown that, under the influence of the forcing signal, some unstable periodic orbits of the chaotic oscillator leave the bulk of the attractor and are visited only very rarely. The investigation of phase synchronization of chaos in terms of unstable periodic orbits has in fact revealed a series of other intriguing features that are not peculiar of periodic oscillations. These features will be described in their full details in later sections of the present chapter, both in the context of phase synchronization transition and imperfect phase synchronization, and in the context of the experimental verification of the scaling properties of phase slips close to the transition point to phase locking.


45

3.1.4. Phase synchronization of coupled chaotic oscillators To complete the discussion on phase synchronization of chaotic oscillators, we now move to describe the way for two chaotic oscillators to lock their phase as a consequence of a weak coupling or interaction. Again, in order to ground our discussions with an illustration, we consider to a pair of coupled chaotic Rössler oscillators [6], which are described by the following equations x˙1,2 = −ω1,2 y1,2 − z1,2 + C(x2,1 − x1,2 ), y˙1,2 = ω1,2 x1,2 + ay1,2 , z˙ 1,2 = f + z1,2 (x1,2 − c).

(3.14)

The two oscillators considered here are slightly nonidentical, in the sense that they display a small parameter mismatch ω1,2 = 0.97 ± ω (the other parameters, a = 0.165, f = 0.2 and c = 10, are the same for the two oscillators). Therefore, though both oscillators display a very coherent phase dynamics due to the proper rotation with a small variation in the return time, they have different average natural frequencies, as a result of the mismatch ω. In Figure 3.8(a) it is shown that, once one fixes the frequency mismatch ω, the two oscillators experience a transition as a function of the coupling strength from a nonsynchronous state (where the phase difference increases almost linearly in time φ1 − φ2 ∼ Ωt), to a synchronous regime where the phase difference is bounded in time (|φ1 − φ2 | < const), and where the difference between the two mean frequencies Ωi = φ˙ i (i = 1, 2) vanishes, i.e., Ω = Ω1 − Ω2 = 0. Once again one has to notice that the above locking condition does not impose any constrain on the chaotic amplitudes of the oscillations, and therefore the phase synchronization regime is not associated with correlations in the two chaotic amplitudes. Figure 3.8(b) indeed reports the trajectory in the subspace (A1 , A2 ) of the phase space (the subspace of the two chaotic amplitudes), and show how such trajectory is far from being distributed close to the diagonal A1 = A2 . This means that the emergence of a phase synchronization regime can be considered as a weaker degree of synchronization in chaotic systems, in contrast to larger degrees of synchronization, such as the complete synchronization phenomenon discussed in Chapter 2. Due to this property, phase synchronization generally occurs already for very low coupling values, and in some cases it can have extremely small thresholds, as can be seen by the synchronization region in the parameter space C ∼ ω in Figure 3.9, which is very similar to the “Arnold tongue” structure of coupled periodic oscillators.

46


Figure 3.8. From Ref. [15]. The phase synchronization regime of two coupled nonidentical Rössler chaotic oscillators. The upper panel reports the time series of the phase difference for different coupling strength C (phases are perfectly synchronized for C > 0.036). The lower panel reports the plot of the amplitude A1 vs. A2 for the phase synchronized case at C = 0.04. Notice that, although the phases are locked, the amplitudes remain chaotic and nearly uncorrelated.

Figure 3.9.

From Ref. [15]. The synchronization region for two coupled Rössler chaotic oscillators in the parameter space (C, ω).

3.2. Transition to phase synchronization of chaos

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3.2. Transition to phase synchronization of chaos After having discussed the properties of phase synchronization of coupled chaotic oscillators, we now move to illustrate the peculiarities of the transition to phase synchronization of chaotic systems, or in other words, how de-synchronization processes can be statistically described when a parameter moves out the phaselocking region. Figure 3.8 shows that, as far as the coupling strength is well outside the synchronization region, the phase difference between the two coupled systems increases almost linearly in time. However, as soon as the parameters approach the border of the phase synchronization region, the temporal behavior of the phase difference displays an alternation between many phase-synchronized epochs (each one of them characterized by a plateau in the plot of the phase difference vs. time) and sudden events (called phase slips) where the phase difference increases its value by 2π. The average duration of the synchronization epochs (called also laminar periods) becomes longer and longer as the parameters approach more and more the Arnold tongue, and phase synchronization is only interrupted intermittently by 2π phase slips, till the system achieve perfect phase-locking inside the synchronization region. It is known for the classical case of coupled periodic oscillators that the transition to phase synchronization corresponds to a saddle-node bifurcation, and the intermittent behavior of phase slips occurring just outside the synchronization region is characterized by the type-I intermittency [74]. This is tantamount to say that the average duration τ1 of the laminar periods (the average time between two successive phase slips) scales as τ1 ∼ |C − Cps |−1/2 ,

(3.15)

with C being a system parameter, and Cps being the transition point of phase synchronization. When we consider a coupling between chaotic oscillators, the situation is a little more complicated, since the synchronization region for a chaotic oscillator corresponds to the intersection (overlap) of the different phase-locking regions of the unstable periodic orbits embedded in the chaotic attractor, and one has, therefore, to take into account explicitly the structure of periodic orbits constituting the skeleton of the chaotic attractor. Precisely, in the phase synchronization region, and for a particular unstable periodic orbit, one has a stable fixed point φs and an unstable fixed point φu , and accordingly, each of the unstable periodic orbit is associated with an attractor and a repeller in the direction of φ. In the generalized phase space (x, φ), the attractor (x, φs ) and the repeller (x, φu ) are well separated, as shown in Figure 3.10(a) for the mapping system

48


Figure 3.10. The stable (in the φ direction) (pluses) and unstable (filled circles) periodic orbits with periods 1-8 forming the skeletons of the attractor and the repeller, respectively. The left panel describes the case of the fully phase-locking region, where the attractor and the repeller are distinct. The right panel reports the case occurring just beyond the border of phase-locking region, where attractor and repeller collide, and the chaotic attractor as a whole is no longer attractive in the φ direction. Reprinted with permission from Ref. [12]. © 1999 The American Physical Society

of equations (3.13). In this generalized phase space with unwrapped phase variable (the phase variable is considered on the real line rather than on the circle, and phase points separated by 2π are not considered as the same), the repellers are periodic orbits on the basin boundary of the attractors [7]. When the parameter cross the phase-locking boundary, the attractor and the repeller of a few of such unstable periodic orbits approach to each other, coalesce and annihilate as a result of the saddle-node bifurcation, as illustrated in Figure 3.10(b). The result of this process is that these unstable periodic orbits are not locked by the external force and phase slips may occur. The dynamics on the weakly unstable direction φ is still the same as in the usual saddle-node bifurcation with a characteristic time between phase slips given by equation (3.15). If the parameters are just beyond the transition point, one has that most of the unstable periodic orbits are still attractive in the φ direction, and 2π slips in the phase difference of the chaotic system can only develop in those portion of time in which the chaotic trajectory is closely visiting one of the unlocked periodic orbits. In other words, to allow at least one slip to occur, the chaotic trajectory should stay for at least a time period τ1 in a close vicinity of an unlocked periodic orbit. Due to the ergodic properties of a chaotic attractor, the probability that the trajectory remains close to a particular unstable periodic orbit for a duration τ1 is proportional to exp(−λτ1 ), being λ the largest Lyapunov exponent of the chaotic system.


49

Obviously, the average time between successive phase slips will be the inverse of such probability, which reads, τ ∼ exp k|C − Cps |−1/2 . (3.16) This scaling law indicates that phase synchronization epochs become extremely long when C approaches Cps , much more than the case of periodic oscillators. One notices immediately that, close to the transition point, the average frequency difference, which is proportional to the inverse of τ , scales as ln |Ω| ∼ −|C − Cps |−1/2 .

(3.17)

3.3. Experimental verification of the transition to phase synchronization In this section we give the due emphasis to the fact that the two above mentioned scaling behaviors of laminar periods close to the bifurcation point to phase synchronization have been experimentally confirmed in a recent study with a forced CO2 laser system [75]. Indeed, being PS the weakest stage of synchronization, and therefore the stage which is first observed experimentally as a coupling parameter is increased, a relevant issue was to fully confirm the transition route to such a behavior from unsynchronized motion. Following what discussed in the previous section, one expects for chaotic systems a scenario in which, if νc is the value of the forcing frequency that marks the transition to PS, the system is phase synchronized for ν < νc . When considering forcing frequencies ν νc , another transition point νt > νc exists such that, for ν > νt the scaling law for τ is the same as the classical case (τ ∼ |ν − νt |−1/2 ), while for νc ν < νt , the intermittency shifts from type-I to that of superlong laminar periods described by 1 (3.18) ∼ −|ν − νc |−1/2 . τ All the above transition scenario has been fully confirmed in a recent experimental study [75]. The experimental setup that was used (Figure 3.11) consists of a CO2 laser tube, pumped by an electric discharge current of 6 mA and inserted within an optical cavity closed by a totally reflecting mirror and a partially reflecting one. The detected laser output intensity suitably amplified drives an intracavity electrooptic modulator that controls the cavity losses. Precisely, the feedback loop is realized by the voltage exiting a HgCdTe fast infrared diode detector, conveyed into an amplifier together with a bias voltage B0 , and driving the electro-optic modulating crystal. ln

50


Figure 3.11. (a) Sketch of the experimental setup. The numbers in the figure denotes the different components of the laser experiment: 1 – Mirrors delimiting the optical cavity. 2 – CO2 laser tube. 3 – Intracavity electro-optic modulator. 4 – HgCdTe fast infrared diode detector. 5 – Amplifier. 6 – Generator for the pumping discharge. 7 – External modulation. 8 – PC based classification of phase slips. (b) Temporal evolution of the laser intensity (in arbitrary units) experimentally observed at zero external amplitude modulation. (c) Synchronization parameter R (see text for definition) vs. ν. The three circles denote the ν values for which phase slips are reported in Figure 3.12. Reprinted with permission from Ref. [75]. © 2002 The American Physical Society


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In these conditions, and in absence of any further modulation, the output intensity consists of a train of homoclinic spikes repeating at chaotic times and interconnected by minor oscillations (see Figure 3.11(b)). In Ref. [75], a square signal modulation was added in the pumping discharge whose amplitude provided a ∼2% perturbation in the electric discharge current, entering a regime of PS by moving the frequency of the external modulation ν, which was applied on a control unit of the generator (element 6 in Figure 3.11(a)). While the phase φe of the external modulation can be straightforwardly evaluated as φe = 2πνt, the phase φs of the chaotic signal is calculated by linear interpolation between successive spiking times φs = 2πk + 2π

t − Tk , Tk+1 − Tk

Tk t < Tk+1 ,

(3.19)

where Tk denotes the time at which the kth spike is produced. Calling R the ratio between the number of maxima in the input modulation and the number of output spikes, Figure 3.11(c) reports the observed route toward PS (R = 1) as ν approaches νc 1.62 kHz, and highlights the process of phase entrainment operated by the external modulation. By means of a PC based acquisition routine, sequences of more than 150,000 inter-spike intervals were recorded, and the occurrences of phase slips in the proximity of the transition point to PS were studied. Figure 3.12 reports the temporal evolution of ≡ |φe − φs | for (a) ν = 2.05 kHz, (b) ν = 1.85 kHz and (c) ν = 1.70 kHz. A sequence of 2π phase slips characterizes the evolution of . Their occurrence becomes rarer and rarer as ν approaches νc . One can furthermore calculate the distribution of inter-slip time intervals (ITI) and monitor its coherence factor τ C≡ , (3.20) σ as a function of ν. Here τ is the average inter-slip time interval, and σ the standard deviation of the ITI distribution. One sees, for ν = νPPS > νc , a maximum in the coherence factor C close to the transition point for PS. Figure 3.13 gives a sketch of the behavior of C(ν). It is apparent the presence of a maximum at νPPS 1.84 kHz, where maximal coherence is produced in the ITI distribution close to the 1:1 phase locking regime. The temporal evolution of at νPPS is shown in Figure 3.12(b), where one can see that phase slips are almost equi-spaced in time. For chaotic systems, the PS region corresponds to the overlap of all the phaselocking regions of the unstable periodic orbits (UPO) embedded in the chaotic attractor [73]. For what said above, one should expect a type-I intermittent scaling law only for frequencies ν > νt , where νt denotes the value for which all UPOs are in the unlocked regime, so as phase slips can occur independently of the particular

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Figure 3.12. Temporal evolution of the phase difference ≡ |φe − φs | for (a) ν = 2.05 kHz, (b) ν = 1.85 kHz and (c) ν = 1.70 kHz. In all cases, 2π phase slips characterize the evolution of , but the occurrence of these slips becomes rarer and rarer as ν approaches the critical value νc . Reprinted with permission from Ref. [75]. © 2002 The American Physical Society

UPO that is visited by the chaotic trajectory. On the contrary, for νc < ν < νt a superlong laminar behavior occurs, since phase slips are allowed only when the chaotic trajectory stays for a sufficiently long time close to those UPOs belonging to the unlocked regime. In order to confirm the above expectations, a series of measurements have been performed at different values of ν, obtaining the results shown in Figure 3.14. The best fits yield νc = 1.62 kHz and νt = 1.84 kHz. Besides confirming the existence of two different scaling behaviors, Figure 3.14 shows that the crossover point for the two scalings coincides with νPPS of Figure 3.13, thus indicating that the coherence between successive phase slips mediates the transition from type-I to super-long laminar period intermittency.


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Figure 3.13. Coherence factor C (see text for definition) vs. external modulation frequency ν. The arrow at νPPS 1.84 kHz indicates the frequency value for which phase slips are maximally coherent. The circles surround the three points for which measurements of (t) are reported in Figure 3.12. In particular, the temporal evolution of at νPPS is shown in Figure 3.12(b), where one can see that phase slips are indeed almost equi-spaced in time. Reprinted with permission from Ref. [75]. © 2002 The American Physical Society

3.4. Ring intermittency at the transition to phase synchronization Recently, it has been pointed out that the transition scenario to phase synchronization of chaotic system is even richer than that described above [76], including another type of intermittent behavior that can be observed near the phase synchronization boundary of two unidirectionally coupled chaotic oscillators. Indeed, up to now, we have mostly considered cases in which the nonidenticity between the coupled systems resulted in a tiny mismatch of the original frequencies. But it is known already for externally driven periodic oscillators that two different scenarios exist for the destruction of synchronization, corresponding respectively to small and large detunings between the oscillator’s frequency and that of the external forcing. For instance, as far as the periodically forced weakly nonlinear isochronous oscillator are concerned, a way to describe the oscillator’s behavior is that of making use of the complex amplitude method in the form u(t) = Re a(t)eiωt , being ω the natural frequency of the oscillator.

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Figure 3.14. The two different scaling behavior of the interslip time intervals observed experimentally close to the transition to phase synchronization: the type-I intermittency scaling behavior (a) and the superlong laminar scaling behavior (b). Dots indicate the experimental measurements. Lines are the best fits τ = −3.4+4.2|ν −νt |−1/2 (νt = 1.84 kHz (a)) and log(1/τ ) = −0.13−0.51|ν −νc |−1/2 (νc = 1.62 kHz (b)). The crossover point for the two scaling laws is located at ν = νt = 1.84 kHz, that corresponds exactly to the value νPPS of maximal coherence in the phase slip occurrence reported in Figure 3.13. Reprinted with permission from Ref. [75]. © 2002 The American Physical Society

Furthermore, for the complex amplitude a(t) of the oscillations, one can write an averaged equation, describing the interaction with the external driving force of the kind a˙ = −iνa + a − |a|2 a − ik, where ν is the frequency mismatch, and k is the (re-normalized) amplitude of the external force.


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If one now imagines to be in a regime of small frequency mismatch (for small values of ν), and if one considers the regime of large k, then the stable solution a(t) = Aeiφ = const corresponds to the synchronous regime, and the destruction of synchronization corresponds to a saddle-node bifurcation on the plane of the complex amplitude. A different scenario, instead, characterizes the regime of large frequency mismatches and decreasing values of k, where the phenomenon of destruction of phase entrainment comes out to be connected with the limit cycle location on the complex amplitude plane. Precisely, as the limit cycle starts enveloping the origin, the synchronization regime begins to destroy. This second scenario can be summarized as follows: if one considers the behavior of the synchronized periodic oscillator on the plane (x , y ) rotating with the frequency of the external signal around the origin, one observes the stable node for the small values of the frequency detuning and a cycle for the large ones, respectively. All these considerations on the rotating plane may easily visualized by using the coordinate transformation x = xr cos ϕd + yr sin ϕd , and y = −xr sin ϕd + yr cos ϕd , being ϕd = ϕd (t) the instantaneous phase of the driving external force. These effects were generalized to the case of chaotic oscillators in Ref. [76], that considered the case of a pair of unidirectionally coupled Rössler systems, whose equations read as x˙d = −ωd yd − zd ,

x˙r = −ωr yr − zr + ε(xd − xr ),

y˙d = ωd xd + ayd ,

y˙r = ωr xr + ayr ,

z˙ d = p + zd (xd − c),

z˙ r = p + zr (xr − c),

(3.21)

where (xd , yd , zd ) [(xr , yr , zr )] are the Cartesian coordinates of the drive (the response) oscillator, dots stand for temporal derivatives, and ε is a parameter ruling the coupling strength. The other control parameters of equation (3.21) have been set to a = 0.15, p = 0.2, c = 10.0. The ωr -parameter (representing the natural frequency of the response system) is ωr = 0.95; the analogous parameter for the drive system is ωd = 1.0. Both chaotic attractors of the drive and response systems are, therefore, phase coherent at zero coupling strength, and phase synchronization occurs around εc ≈ 0.124. Due to phase coherence of the two chaotic oscillators, the instantaneous phase ϕ(t) can be therefore introduced in the traditional way, as the rotation angle

56


ϕd,r = arctan(yd,r /xd,r ) on the projection plane (x, y) of each system, and phase synchronization can be detected by monitoring the time evolution of the instantaneous phase difference, that has to obey the phase locking condition ϕ(t) = ϕd (t) − ϕr (t) < const. (3.22) Ref. [76], considered the case of large parameter mismatches (e.g., a large difference in the natural frequencies of the drive and response systems), and described the occurrence of a different kind of intermittent behavior emerges, that was called ring intermittency. In analogy with the scenario described for forced periodic systems, Figure 3.15(a) reports the behavior of the synchronized response oscillator (3.21) on the plane (x , y ) rotating around the origin in accordance with the phase ϕd (t) of the drive system when the control parameters ωd and ωr are detuned sufficiently.

Figure 3.15. The phase trajectories of the response system on the (x , y )-plane rotating around the origin. Panel (a): the phase synchronization regime occurring at ε = 0.126; panel (b): the ring intermittency regime for ε = 0.115; panel (c): the asynchronous dynamics occurring at ε = 0.109, for which the phase trajectory envelops the origin of the plane at each rotation. Panel (d) reports the corresponding temporal evolutions of the phase difference ϕ(t). Reprinted with permission from Ref. [76]. © 2006 The American Physical Society


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One can see that the phase trajectory on this plane looks like a ring, consisting of the chaotic phase trajectories, instead of the limit cycle that would occur in the periodic case. When the coupling strength ε gets below the critical value εc the phase trajectory on the (x , y )-plane starts enveloping the origin (see Figure 3.15(b), the origin is the point of intersection of the dashed lines), and the phase synchronization regime begins to destroy, as phase slips begin to be observed all the times that the phase trajectory envelops the origin of that plane. As the coupling strength decreases further, the phase trajectory envelops the origin more often, and the phase slips occur more frequently. Eventually, as the coupling strength ε becomes less than εt ≈ 0.1097, the origin is inside the ring (see Figure 3.15(c)), therefore every rotation of phase trajectory causes a phase slip. Summarizing the observed scenario one can see that, as the coupling strength ε is progressively decreased, the following dynamical behaviors are produced: (i) the phase synchronization regime for ε > εc , (ii) the intermittent behavior for εt < ε < εc and (iii) the asynchronous dynamics for ε < εt when the phase slips follow each other at approximately equal time intervals T , the averaged period of the phase trajectory rotation on the (x , y )-plane (see Figure 3.15(d)). Ref. [76] further analyzed the observed intermittent behavior, and extracted the scaling properties characterizing the dependence on the coupling strength of the duration and distribution of laminar periods. In order to make this latter analysis, Ref. [76] started with considering the probability that the phase trajectory on the rotating (x , y )-plane envelops the origin be p = p(ε). One has that p = 0 for ε > εc , p = 1 for ε < εt and 0 < p < 1 when εt < ε < εc . In the case of the intermittent behavior (i.e., εt < ε < εc ) the probability of the laminar phase with length T to be observed is P (T ) = p 2 . This period is determined by the difference of the main frequencies of the drive (fd ) and response (fr ) systems and may be calculated as T ≈ 1/|fr − fd |. For the parameters mentioned above, one has that T ≈ 80. Furthermore, the probability of laminar phases with length nT to arise is P (nT ) = (1 − p)n−1 p 2 . Therefore, the distribution of the laminar phases with generic length τ should scale as N(τ ) ∼ p2 (1 − p)τ/T −1 ,

τ > T.

(3.23)


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Equation (3.23) may be also rewritten in the form N(τ ) = A exp(kτ ),

τ > T,

(3.24)

where k = (1/T ) ln(1 − p), A being a normalizing coefficient. Thus, the laminar phase distribution in the ring intermittency regime obeys an exponential law, with the parameter k being negative due to 0 < p < 1. Following again Ref. [76], one can derive explicitly the dependence of the mean length τ of the laminar phases (i.e., the averaged time interval between two successive phase slips) on the coupling strength ε. The relationship between the mean length τ (ε) and the probability p = p(ε) is τ = T −

1 T =T − . k ln(1 − p)

(3.25)

One numerically observes that, in the coupling strength range εt < ε < εc , the value of the probability p is directly proportional to the deviation of the coupling strength ε from the critical value εc [76], i.e., p(ε) ∼ (εc − ε).

(3.26)

This is confirmed in Figure 3.16, where it is reported the dependence of the probability p on the deviation of the coupling strength from the critical value (εc − ε). Since the probability p on the coupling strength in the range εt < ε < εc relates to the coupling strength as p(ε) = (εc − ε)/(εc − εt ), one easily obtains that the dependence of the mean laminar phase length on ε has to scale in the form −1 ε − εt τ (ε) = T 1 − ln (3.27) . εc − εt It has to be remarked that equation (3.27) describe a scaling behavior which is substantially different from that of the super long laminar periods, and therefore one has to conclude that the initial frequency mismatch is a relevant parameter in setting up the scaling of phase slips’ occurrence in the proximity of the Arnold tongue.

3.5. Imperfect phase synchronization Up to now, we have basically limited the description of phase synchronization phenomena to the case of phase coherent chaotic oscillators. However, since phase synchronization corresponds to an entrainment process of the time scales of one

3.5. Imperfect phase synchronization

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Figure 3.16. Plot of the probability p that the phase trajectory on the rotating (x , y )-plane envelops the origin as a function of the deviation of the coupling strength from the critical value (εc − ε). The numerically calculated points are here reported in symbols (♦), while the linear approximation p = a(εc − ε) (where a = 70) is depicted as a solid line. The critical values εc and εt for the coupling parameter, delimiting the range of occurrence of the ring intermittency phenomenon, are shown by arrows. Reprinted with permission from Ref. [76]. © 2006 The American Physical Society

oscillator to those of the other, it is to be expected that also the distribution of time scales in the original oscillators plays actually a crucial role in the synchronization process. And, indeed, this is the case for phase noncoherent oscillators, for which the corresponding power spectrum is not peaked around a well defined mean frequency, but it is dispersed on a wide range of frequencies. In the viewpoint of unstable periodic orbits, phase coherence of a chaotic oscillator means that the orbits have narrowly distributed periods, and consequently, the phase-locking regions corresponding to such orbits do not differ substantially from one another, warranting the existence of a nonnull overlapping region that would correspond to full phase synchronization of the chaotic attractor. However, when the system has a rather broad distribution of time scales for periodic orbits, an external signal with a given frequency may not be able to entrain all of such time scales at once, and therefore phase slips would unavoidably occur when the chaotic flow comes to oscillate with time scales outside the synchronization region of the driving signal. This behavior has been called imperfect phase synchronization [12,77,78]. In order to capture the difference between the phase coherent and phase noncoherent case, we follow the approach of Ref. [12,77,78], and carry out a study


60

of perfect and imperfect phase synchronization with a periodically driven Lorenz system, x˙ = 10(y − x), y˙ = rx − y − xz, z˙ = xy − 2.667z + E cos(Ωt).

(3.28)

This system, indeed, exhibits a rich bifurcation scenario, as the parameter r is varied [27]. At r = 210, the chaotic oscillations result from a periodic-doubling route, and the system can be synchronized perfectly by a periodic driving signal with a frequency close to the average frequency Ω = 24.92, as it can be seen by the plateau of the vanishing frequency difference ω − Ω = 0 in Figure 3.17. The situation becomes different for r = 28, where a certain plateau of ω −Ω = 0 appears; however, this plateau is neither horizontal nor lies at zero. As a result, phase synchronization in this case is not perfect, as shown by 2π or 4π phase slips in Figure 3.18. The reason for such imperfect phase synchronization is that, at r = 28, there is a saddle point (0, 0, 0) embedded in the chaotic attractor. The trajectory therefore experiences a considerable slowing down when it comes close the saddle point, while the oscillations are much quicker as the trajectory rotates around either one of the two unstable foci [27]. In correspondence of this behavior, the time scales have a relatively large variation around the average value, as can be appreciated

Figure 3.17. Perfect and imperfect phase synchronization for the Lorenz system. At r = 210 and E = 3, the system can be synchronized perfectly by a periodic driving signal with a frequency close to the average frequency Ω = 24.92, as it can be seen by the plateau of the vanishing frequency difference of the dotted curve. At variance, for r = 28 and E = 6 the system cannot achieve perfect phase synchronization and the frequency difference (now the solid line) does not feature any plateau around zero. Reprinted with permission from Ref. [12]. © 1999 The American Physical Society

3.6. Lag synchronization of chaotic oscillators

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Figure 3.18. Temporal evolution of the phase difference in the case of imperfect phase synchronization. Notice the 2π or 4π phase slips that persistently occur in time. The inset is a zoom reporting the behavior of the phase difference in the proximity of one of such phase slips. Reprinted with permission from Ref. [12]. © 1999 The American Physical Society

by looking at by the distribution of the frequency of different unstable periodic orbits in Figure 3.19. On its turn, this large variations in frequency makes no possible the existence of an overlapping region for the Arnold tongues of the different unstable periodic orbits, and therefore a full synchronization region of the chaotic attractor does not exist: for each given frequency and amplitude of the external driving signal, there exist always certain unstable orbits which are not locked, and when the chaotic trajectory visits closely any one of them, phase slips are unavoidably produced in the temporal behavior of the phase difference. The full description of this interesting phenomenon can be obtained in Refs. [12,77,78].

3.6. Lag synchronization of chaotic oscillators In the previous sections, we have concentrated our attention to the description of the phenomenon of phase synchronization. As one moves toward larger values of the coupling strength, it is reasonable to expect that some kind of relationships would begin to be established also between the amplitudes.


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Figure 3.19. Individual frequencies of unstable periodic orbits embedded into the Lorenz attractor at r = 28. The dashed line represents the mean frequency of the autonomous chaotic motion. Notice that the time scales associated with the different unstable periodic orbits have a relatively large variation around the average value. On its turn, this makes no possible the existence of an overlapping region for the Arnold tongues of the different orbits, and therefore a full synchronization region of the chaotic attractor does not exist. Reprinted with permission from Ref. [12]. © 1999 The American Physical Society

And indeed, it has been shown in Ref. [8] that the phase synchronization regime is followed by another synchronization regime that has been called lag synchronization. This latter synchronization regime corresponds to the case in which the states of the two oscillators become nearly identical, but one system is shifted in time with respect to the other. In order to quantitatively characterize this new phenomenon, Ref. [8] introduced the similarity function S(τ ), which corresponds to the time averaged difference between the variables x1 (t) and x2 (t + τ ) (with mean values being dropped) S 2 (τ ) =

(x2 (t + τ ) − x1 (t))2 . x12 (t)x22 (t)

(3.29)

This measure is similar to the cross correlation function (x2 (t + τ )x1 (t))2 . But S is especially suitable for measuring lag synchronization from bivariate time series, because S(τ0 ) ≈ 0 at a certain nonzero τ0 indicates the setting of a lag synchronization regime. The value of τ for which the similarity function displays the minimum individuates the time shift between the temporal sequences of the two oscillators. A typical feature of S for the coupled Rössler oscillators are shown in Figure 3.20 for different values of coupling C. When the system is in the phase


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Figure 3.20. From Ref. [15]. Similarity function obtained for two coupled Rössler oscillators (equation (3.14)) for different values of the coupling strength C. In the phase synchronization regime (e.g., C = 0.05), the curve has a clear minimum σ = 0. Lag synchronization occurs when σ becomes effectively zero (e.g., C = 0.15).

synchronization regime, a minimum can already be seen clearly, but the corresponding value of the similarity function is well above zero, because even though the phases are locked, there is no sufficient correlation between the chaotic amplitudes. With increasing C both τ0 and S decrease. When C approaches Clag = 0.14 where S effectively reaches zero at a nonzero τ0 , the system undergoes a transition to lag synchronization. After identifying the time delay τ0 by this similarity function S, the lag synchronization can be graphically visualized by reporting the plots of x(t + τ0 ) vs. x(t) which will be restricted to an almost straight line. Furthermore, Ref. [79] has shown experimentally that lag synchronization is robust to perturbations to some extend.

3.7. Intermittent lag synchronization In order to describe more quantitatively the lag synchronization phenomenon, as well as the transition from phase to lag synchronization, we here follow the discussion of Ref. [11] that focused on the intermittent lag synchronization, a phenomenon that occurs in between phase and lag synchronization and is characn (n = 1, 2, . . .), such that terized in terms of the existence of a set of lag times τlag n ) for a given n. the system always verifies x1 (t) x2 (t − τlag In order to proceed, we will refer to a pair of coupled nonidentical Rössler systems, describing the evolution of the three dimensional vectors x1,2 ≡ (x1,2 , y1,2 , z1,2 ):

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Figure 3.21. The typical time series of x2 (t) − x1 (t − τ0 ) characterizing the intermittent lag synchronization regime. The case reported in the figure refers to ε = 0.13, τ0 = 0.32. Notice that, most of the time the system verifies |x2 (t) − x1 (t − τ0 )| 1, but bursts of local nonsynchronous behavior persistently and intermittently occur in time. Reprinted with permission from Ref. [11]. © 2000 The American Physical Society

x˙1,2 = −ω1,2 y1,2 + ε(x2,1 − x1,2 ), y˙1,2 = ω1,2 x1,2 + ay1,2 ,

(3.30)

z˙ 1,2 = f + z1,2 (x1,2 − c), where dots denote temporal derivatives, a = 0.165, f = 0.2, c = 10 so as equations (3.30) generate a chaotic dynamics, ε represents the coupling strength, and ω1,2 ≡ ω0 ± ( being the frequency mismatch between the two chaotic oscillators). In what follows we focus our study on the case ω0 = 0.97, = 0.02. As ε increases, Ref. [8] identifies subsequent transitions in system (3.30) from no synchronization to phase, to lag and to complete synchronization, and traces each one of these transitions in the Lyapunov spectrum. In particular, in the range 0.11 < ε < 0.14 (that is in between perfect phase and perfect lag synchronization), Ref. [8] describes that is a situation where most of the time the system verifies |x2 (t) − x1 (t − τ0 )| 1 (τ0 being a lag time), but where bursts of local nonsynchronous behavior may occur. In this range of coupling strengths, the typical output of system (3.30) is reported in Figure 3.21. Let us again refer to the similarity function S(τ ) of equation (3.29). If the similarity function shows a global minimum σ = minτ S(τ ), for τ0 = 0, this is an indication of the presence of a principal lag time τ0 between the two processes.


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Figure 3.22. Upper panel: The similarity function S 2 (τ ) vs. the lag time τ for ε = 0.13 (i.e., in the regime of intermittent lag synchronization). Notice the presence of a global minimum σ 0 at τ0 = 0.32, and of many other local minima for larger lag times τn (n = 1, 2, 3, . . .). The lower panel is a zoom of the region close to the origin. Solid line: ε = 0.13, dashed line: ε = 0.1, dotted line: ε = 0.07. Reprinted with permission from Ref. [11]. © 2000 The American Physical Society

In Figure 3.22(a), we report the similarity function for ε = 0.13, that is within intermittent lag synchronization. Looking at Figure 3.22(a), one clearly realizes that, besides a global minimum σ 0 at τ0 = 0.32, S(τ ) displays many other local minima at larger lag times τn (n = 1, 2, 3, . . .). These local minima witness that system (3.30), besides being lag synchronized most of the time with respect to the global minimum τ0 , during its dynamical evolution occasionally visits closely other lag configurations corresponding to |x2 (t) − x1 (t − τn )| 1. The deepness of the nth local minimum is in close relationship with the fraction of time that the corresponding lag configuration is closely visited by system (3.30).

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Both the location and the deepness of all local minima are functions of the coupling strength ε. Figure 3.22(b) report a zoom of the similarity function (limited to the first two local minima) at different values of ε. As the coupling strength increases, two different phenomena can be observed in Figure 3.22(b), namely a drift of all τn toward the origin, and a change in their corresponding deepnesses. Precisely, the value of the local minima τn increases, while the value of the global minimum τ0 decreases, vanishing eventually at larger ε values. The decreasing process of τ0 reveals that, increasing the coupling strength, the system will eventually lead to a complete synchronization state. The increasing process of the values of the local minima tells us that one is approaching a perfect lag synchronization state (wherein the fraction of time spent by system (3.30) in the configuration |x2 (t) − x1 (t − τ0 )| 1 must increase, and consequently the fraction of time spent in all configurations |x2 (t)−x1 (t −τn )| 1, n = 0 must vanish). What reported in Figure 3.22 suggests to look carefully not only to lag synchronization phenomena with respect to the principal lag time τ0 , but also to the occurrence of the other lag configurations with n > 0 during intermittent lag synchronization. To this purpose, Ref. [11] selected ε = 0.13 in system (3.30) and monitored the differences n = x2 (t) − x1 (t − τn ),

n = 0, 1, 2, 3

(for this particular choice of ε, τ0 = 0.32, τ1 = 6.60, τ2 = 12.87, and τ3 = 19.20). The results are shown in Figure 3.23. For most of the time the oscillation of 0 are strongly bounded. Correspondingly the oscillations of n are very large for all n. However, during the burst of local nonsynchronous behavior (with respect to the lag time τ0 ), the oscillations of 1 and 3 range within a limited interval. This means that, during the intermittent bursts, the system visits closely another lag configuration corresponding to some lag time τn . In other words, one can interpret intermittent lag synchronization as the coexistence of many lag configurations (each one of them corresponding to a different local minimum of the similarity function) and the intermittency phenomena can be regarded as erratic jumps from one to another of such configurations. In order to better visualize what is happening, Ref. [11] compared the dynamics of system (3.30) inside and outside the intermittent bursts. In Figure 3.24(a) it is shown a snapshot of the dynamical evolution of 0 for ε = 0.13 around the occurrence of an intermittent burst. Two different temporal regions (denoted by arrows in Figure 3.24(a)) are selected, namely a region inside and one outside the burst. The corresponding portions of the chaotic trajectory can be visualized in the subspaces (x1 , y1 ) and (x2 , y2 ) (Figure 3.24(b) within the burst, and Figure 3.24(c) outside the burst).


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Figure 3.23. Temporal behavior of the differences n = x2 (t) − x1 (t − τn ), for n = 0 (panel (a)), n = 1 (panel (b)), n = 2 (panel (c)) and n = 3 (panel (d)). In all cases ε = 0.13. (a) τ0 = 0.32; (b) τ1 = 6.60; (c) τ2 = 12.87; (d) τ3 = 19.20. It is evident that the intermittent lag synchronization can be interpreted as the coexistence of many lag configurations (each one of them corresponding to a different local minimum of the similarity function) and the intermittency phenomena can be regarded as erratic jumps from one to another of such configurations. Reprinted with permission from Ref. [11]. © 2000 The American Physical Society

The two trajectory portions have been selected so as to contain the same number of oscillations (around 40), in order to qualitatively highlight the relative differences between the two cases. While outside the burst the trajectory is spread within the chaotic attractor (Figure 3.24(c)), inside the burst one can clearly appreciate the multibranch structure closely visited by the chaotic trajectory, revealing proximity to a period 2 orbit.

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Figure 3.24. (a) Temporal evolution of 0 during the occurrence of an intermittent burst (ε = 0.13); (b) and (c): projections of the portion of the dynamical evolution (corresponding to about 40 oscillations, and marked with arrows in (a)) of the two coupled systems onto the corresponding (x, y) planes within (b) and outside (c) the intermittent burst. Notice that, while outside the burst the trajectory is spread within the chaotic attractor, inside the burst the trajectory reveals proximity to a period 2 orbit. Reprinted with permission from Ref. [11]. © 2000 The American Physical Society


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The general conclusion of Ref. [11] is then that intermittent lag synchronization arises as the coexistence of a variety of lag times, and the passage from one to another of the corresponding lag synchronized states occurs in correspondence of the passage of the chaotic trajectory in the proximity of some unstable periodic orbit. The effect of an increasing of the coupling strength is that the principal lag configuration becomes more and more stable, and the consequence is that the system is spending more and more time in the principal lag configuration. Finally, the further increase of the coupling strength determines the value of the principal lag time to progressively vanish, thus inducing a regime similar to complete synchronization.

3.8. Phase synchronization of nonautonomous chaotic oscillators In order to conclude the description of phase synchronization, a further point should be clarified. As it has been described so far, phase synchronization appears to be a phenomenon intimately related to the presence of two distinct self sustained oscillators whose original different rhythms are adjusted by the coupling. As so, one would be tempted to conclude that this phenomenon is mostly limited to the case of autonomous oscillators. In this section we show, however, that phase synchronization can be established also for nonautonomous systems, though it emerges via a completely different scenario. Indeed, since all zero Lyapunov exponents of nonautonomous systems are insensitive to the coupling, phase synchronization here is associated to a distinct dynamical mechanism through which a previously positive Lyapunov exponent vanishes over a broad range of the coupling strength parameter, thus indicating that the rhythm adjustment process here takes place also in the absence of a contractive direction for the phases. To illustrate the phenomenon, we will follow the approach of Ref. [80], and refer to a pair of forced Van der Pol oscillators 2 3 ) + Bx1,2 = C sin(ω1,2 t) x¨1,2 − Ax˙1,2 (1 − x1,2

in a bidirectional symmetrical coupling configuration. The equations of motion read x˙1,2 = y1,2 , 2 ) − Bx 3 + C sin(ω z) + (x y˙1,2 = A1,2 y1,2 (1 − x1,2 1,2 2,1 − x1,2 ), 1,2

(3.31)

z˙ = 1, where the subscripts 1 and 2 refer to oscillator 1 and 2, respectively, dots denote temporal derivatives, A1 = 0.6, A2 = 0.2, B = 1, C = 2, ω1 = 0.6 and

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Figure 3.25. Projections in the plane (y1 , x1 ) of the attractor of equations (3.31) for (a) = 0 (uncoupled case); (b) = 1 (intermediate coupling) and (c) = 2.7 (strong coupling). Reprinted with permission from Ref. [80]. © 2004 The American Physical Society

ω2 = 0.65 are parameters chosen so as to produce a chaotic dynamics for both uncoupled oscillators, and represents the coupling strength. In the uncoupled case ( = 0) and for the selected values of parameters, the two oscillators exhibit a chaotic motion developing onto an attractor which does not display a unique center of rotation (see Figure 3.25(a)). At weak coupling it is to be expected that the effect of the two distinct forcing frequencies ω1 and ω2 is that of hampering frequency and phase synchronization, insofar as the two systems will have a strong component of these frequencies in their Fourier spectra. Intermediate couplings ( = 1 in Figure 3.25(b)) produce a slightly distorted attractor in phase space, which however does not significantly changes the qualitative features of the original chaotic motion. Finally, a stronger coupling ( = 2.7 in Figure 3.25(c)) has the effect of destroying the chaotic attractor and transforming it into a quasi periodic one where


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Figure 3.26. Upper panel: the Lyapunov exponents of equations (3.31) as functions of the coupling parameter . Lower panel: the linear cross-correlation (Pearson’s) coefficient between x1 and x2 as a function of . Notice that, in the intermediate coupling regime 0.56 < < 1.61, the correlation coefficient takes a nearly constant negative value that differs from zero but is not close to −1, giving a signature that some sort of synchronized motion is established, which however differs from complete synchronization. Reprinted with permission from Ref. [80]. © 2004 The American Physical Society

the coupling induced suppression of chaos [81] is associated with the signals x1 (t) and x2 (t) being in anti-phase. As a preliminary step, Ref. [80] started the analysis by making use of the standard tools for the evaluation of Lyapunov exponents in equations (3.31) [82], with the aim of investigating how the Lyapunov spectrum changes as the coupling increases. The results are reported in Figure 3.26(a), where the vertical dashed lines are used as a guide for a better visualization of the different regimes that will be discussed here below. At = 0, the spectrum is composed by two positive, two negative and one zero exponents. This latter one corresponds to the equation z˙ = 1 (and therefore is insensitive to the coupling), accounting for the invariance of equations (3.31) with respect to time translations.


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As increases, one of the originally positive exponents decreases to a slightly negative value in the range 0.3 < < 0.56. For 0.56 < < 1.61, such an exponent remains zero, eventually assuming a negative value for > 1.61. Finally, for > 2.4 no positive Lyapunov exponents are present in the spectrum, indicating that chaos has been suppressed and the signals x1 (t) and x2 (t) are sitting on a quasi-periodic attractor. Further insight on the synchronization process can be obtained by comparing Figure 3.26(a) with the evolution of the linear cross-correlation coefficient between the two temporal series x1 (t) and x2 (t), given by ρ=

(x1 − x1 )(x2 − x2 ) , (x1 − x1 )2 (x2 − x2 )2

(3.32)

and reported in Figure 3.26(b). It is seen that for 0.56 < < 1.61, the correlation coefficient takes a nearly constant negative value that differs from zero but is not close to −1, and therefore there is a signature that some sort of synchronized motion is established, which however differs from complete synchronization. In the quasi-periodic regime ( > 2.4) the value of the correlation coefficient tends to −1, which means that we have complete synchronization in anti-phase. In the following we will concentrate our attention on the intermediate coupling regime 0.56 < < 1.61 and show that it corresponds to a phase synchronization state. A practical difficulty in the analysis is that in the range of couplings 0.56 < < 1.61, each oscillator has not a unique center of rotation, and therefore the standard tools introduced in the case of phase coherent systems do not here help to suitably measure the corresponding instantaneous phases. A first hint of what is happening is given, however, by the inspection of the power spectra of x1 and x2 . Figure 3.27 reports the power spectra of x1 (black) and x2 (gray) for = 0 (uncoupled case (a)), = 1 (in the middle of the phase synchronization range (b)) and = 1.7 (at the border of the phase synchronization range (c)). In all cases, the Fourier spectra are broad band, indicating a phase noncoherent chaotic dynamics, and contain two distinct peaks in correspondence of the two external forcing frequencies ω1 = 0.6 and ω1 = 0.65. In addition, the uncoupled spectra (Figure 3.27(a)) show the presence of two extra peaks, that are located in different frequency positions for x1 and x2 . As one enters the phase synchronization range, the peaks corresponding to the forcing frequencies do not overlap, but a higher peak around ω = 2.1 is set common to the two spectra, where frequency entrainment is obtained (Figure 3.27(b)). The frequency location of this synchronization peak increases approximately linearly with . Finally, for = 1.7, Figure 3.27(c) shows two “synchronization” peaks at ω = 1.6 and ω = 2.6. For larger values, the chaotic attractor becomes structurally unstable.


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Figure 3.27. Power spectrum (in arbitrary units) of the signals x1 (black) and x2 (gray) vs. the frequency ω for (a) = 0; (b) = 1 and (c) = 1.7. Notice that, in all cases, the Fourier spectra are broad band, indicating a phase noncoherent chaotic dynamics, and contain two distinct peaks in correspondence of the two external forcing frequencies ω1 = 0.6 and ω1 = 0.65. The uncoupled spectra (panel (a)) show the presence of two extra peaks, that are located in different frequency positions for x1 and x2 . As one enters the phase synchronization range (panel (b)), the peaks corresponding to the forcing frequencies do not overlap, but a higher peak around ω = 2.1 is set common to the two spectra, where frequency entrainment is obtained. Finally, for = 1.7 (panel (c)), one can appreciate the presence of two “synchronization” peaks at ω = 1.6 and ω = 2.6. Reprinted with permission from Ref. [80]. © 2004 The American Physical Society

This suggested the use of a band-pass filter to properly isolate a filtered signal around the second frequency peak in the Fourier spectra, to which the standard analytic continuation technique by means of the Hilbert transform can be applied for the evaluation of the instantaneous phases. Accordingly, Ref. [80] introduced at each value a pass-band Butterworth filter centered around the second frequency peak in the Fourier spectrum, and, after the filtering process, calculated the instantaneous phases φ1 and φ2 of the two filtered signals. Figure 3.28 reports a long snapshot of the temporal evolution of the instantaneous phase difference φ(t) = φ1 (t) − φ2 (t) outside ( = 0.3) and inside ( = 1) the range for which the second Lyapunov exponent vanishes. Looking at Figure 3.28, one easily realizes that φ diffuses in an almost random fashion toward infinity at = 0.3. At variance, for = 1, φ behaves

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Figure 3.28. Snapshot of the temporal evolution of the instantaneous phase difference φ (see text for definition) for = 0.3 (diverging curve, outside the range for which the second Lyapunov exponent of the system vanishes) and = 1 (bounded curve, inside the range for which the second Lyapunov exponent vanishes). Reprinted with permission from Ref. [80]. © 2004 The American Physical Society

alternating long epochs of almost constant value, interrupted by 2 π jumps (or phase slips), with no apparent trend, thus indicating that the two oscillators are most of the time phase synchronized to each other. The importance of the study performed in Ref. [80] is that of having extended the concept of phase synchronization to the case of nonautonomous chaotic oscillators, and to have pointed out the intrinsic differences existing with the case of autonomous systems.

3.9. Generalized synchronization The next step is to describe the synchronization phenomena emerging between nonidentical systems for a large enough coupling strength, that eventually leads to the emergence of specific functional relationships between the outputs of the coupled systems. In general, indeed, when there exists an essential difference between the coupled dynamical units, the manifold in the phase space attracting the system trajectories can be complicated, and one has to expect that its functional representation


75

would differ substantially from the identity (that would represent the case of complete synchronization between identical systems). Two central points have to be discussed in details, and constitute the fundamental properties of the subject. The first is that one should generalize the concept of synchronization to include nonidenticity between the coupled systems. The second is that, since in general one will obtain complicated functional relationships, not easy to be detected with the standard methods discussed so far, one should design suitable new tools to detect these relationships. Many early works have shown that another type of chaotic synchronization can be set between nonidentical systems [4,9], and have called this phenomenon Generalized Synchronization. In most cases, the evidence of it has been intimately related to a unidirectional coupling scheme. In order to define Generalized Synchronization for an unidirectional coupling scheme, let us be guided by the following example of coupled systems x˙ = F (x), y˙ = G y, hμ (x)

(3.33)

where x is the n-dimensional state vector of the driver and y is the m-dimensional state vector of the response. F and G are vector fields, F : R n → R n , and G : R m → R m . The coupling between response and driver is ruled by the vector field hμ (x) : R n → R m , where the dependence of this function upon the parameters μ is explicitly considered. When μ = 0, the response system evolves independently on the driver, and both systems are chaotic. The existent literature reports some slight differences in the definition of generalized synchronization. We here will follow the most general one, that was given in Refs. [9,10,83]. Such a definition states that, when μ = 0, the chaotic trajectories of the two systems are said to be synchronized in a generalized sense if there exists a transformation φ:x → y which is able to map asymptotically the trajectories of the driver attractor into the ones of the response attractor y(t) = φ x(t) , regardless on the initial conditions in the basin of the synchronization manifold M = (x, y): y = φ(x) . The difference in the definitions that historically were given to generalized synchronization is essentially concentrated on the properties requested for the map φ. In particular, some authors introduced this phenomenon assuming the map φ to


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be homomorphic [4], whereas later the definition was extended also to the case of diffeomorphic maps [84]. Furthermore, in Ref. [19] there is an interesting distinction between two types of generalized synchronization, namely the so-called Weak Synchronization (WS) and the Strong Synchronization (SS), depending on the smoothness or differentiability of the map φ: SS (WS) corresponds to the case of a map φ which is (which is not) smooth, in the sense of being (not being) differentiable. An even stronger condition for generalized synchronization was put in Ref. [85], which introduced the so-called differentiable generalized synchronization, requiring a continuous differentiability for the map φ. It is important to point out that all of these different approaches have relevant consequences at the moment of looking for the existence of generalized synchronization in experimental data, as we will outline momentarily. The necessary and sufficient conditions for the occurrence of generalized synchronization in the system (3.33) were postulated by Kocarev and Parlitz [10]. As in the case of complete synchronization, the notion of generalized synchronization is equivalent to the asymptotic stability of the response system. Let us recall that the response system is said to be asymptotically stable if lim y t, x(0), y1 (0) − y t, x(0), y2 (0) = 0, t→∞

where (x(0), y1 (0)) and (x(0), y2 (0)) are two generic initial conditions of system (3.33). In other words, a map y = φ(x) (not necessarily differentiable) exists whenever the action of the driver results in the response forgetting its initial condition. However, as y = φ(x) may be far from being the identity, it is in general very difficult to unravel generalized synchronization states by a simple inspection of the output signals. The stability of the manifold M of generalized synchronization can be determined as in the case of complete synchronization, i.e., by the negativity of conditional Lyapunov exponents [19], the use of Lyapunov functions [10] and a further criterion given in Ref. [86]. In addition to the problem of stability, other issues related with the existence of generalized synchronization are crucial, especially when trying to detect it in experiments, where one is often interested in discerning whether some synchronization features appear. To this purpose, some useful statistical parameters have been introduced to test the existence of the map φ and to determine its mathematical properties. One of these approaches is given in Ref. [9], which defines a mutual false neighbors parameter r signaling the existence of a one to one smooth map between two dynamical variables. The definition of this parameter is based on the condition that the close points on the reconstructed space of the driver dynamics should


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have close images in the reconstructed space of the response dynamics, allowing to establish the continuity of φ, and thus inferring the existence of a GS state. The method consists in considering the two scalar time series x1 (t) and x2 (t) (the outputs of the two coupled systems), and to construct three different embedding spaces, namely S1 , S2 and S3 . Precisely, S1 is the embedding space of x1 (t) with fixed embedding dimension m1 ; S2 is the embedding space of x2 (t) at variable embedding dimension m2 ; and S3 is the embedding space of x2 (t) at the fixed embedding dimension m1 . The idea is to choose randomly n state vectors xn1 in the space S1 and to consider the images xn2 and xn3 in the spaces S2 and S3 . By images one means here those state vectors in S2 and S3 that correspond to the same time occurrence of xn1 in S1 . Now, at each one of such n state vectors, it is possible to associate xn1,N N1 n (x3,N N3 ), defined as the nearest neighbor to xn1 (xn3 ) in S1 (S3 ), i.e., that vector state—realized at another time in the system—that is the closest to xn1 (xn3 ) in S1 (S3 ). In the very same way, the nearest neighbor xn2,N N2 to xn2 in S2 is considered. Now, by calling xn1,N N2 (xn3,N N2 ) the image of xn2,N N2 in S1 (S3 ), the MFNN parameter is defined as [9,87] r=
n ,

(3.34)

where < ... >n denotes the ensemble average over the selected n states. r ≡ 1 means that neighbor states of one system would correspond to neighbor states of the other, thus implying the settings of a functional relationship between the two reconstructed subsystems [9,87]. Therefore r 1 could be taken as a good indicator for the setting of a generalized synchronization state, whereas 0 r 1 indicates that the system is evolving in an unsynchronized state. A further method to characterize dynamical interdependence among nonlinear systems based on mutual nonlinear prediction is given in Ref. [88]. This method provides information on the directionality of the coupling, and therefore it can be used to detect GS between dynamical variables. This technique was applied to detect GS in a neuronal ensemble. Based on the equivalence between generalized synchronization and the asymptotic stability of the response in system (3.33), a useful physical criterion have been suggested to detect it in an experiment [19,83]. The method consists in constructing an auxiliary system (a replica of the response system), and driving both the response and the auxiliary system by the same driving system y˙ = G y , hμ (x) .

(3.35)


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In this framework, y(t) is said to be synchronized with x(t) in a generalized sense if lim y (t) − y(t) = 0. t→∞

This is tantamount to state that generalized synchronization between x(t) and y(t) occurs if complete synchronization takes place between y (t) and y(t). The main advantage of this criterion is the easy detection of CS between y(t) and y (t), and, indeed, this method has been recently used to experimentally demonstrate that generalized synchronization arises in unidirectionally coupled lasers [89].

3.10. A mathematical definition of synchronization Along this and the former chapters, we have discussed different types of synchronized motions for coupled identical and nonidentical systems. The above plethora of theoretical studies as well as the large number of experimental verifications have historically stimulated the attention of several groups with the goal of establishing a unifying framework for synchronization of coupled dynamical systems. In the course of the last decades, several attempts have been put forward in that direction, and we will refer here to the most recent ones. The first attempt to provide a general definition of synchronized motion was made in Ref. [90]. More recently, Brown and Kocarev [91] proposed a mathematical definition, that assumes to have a system divisible into two subsystems in which functions (or properties) can be defined, consisting in mappings from the space of trajectories and time to some Cartesian space. Formally speaking, this implies that the total system is seen as w = [x, y], w ∈ R m1 +m2 , x ∈ R m1 , y ∈ R m2 , with each subsystem forming a trajectory φx (w0 ) and φy (w0 ) (w0 being a given initial condition). The two trajectories are then mapped by the properties gx and gy to a new space R d . Finally, in order to define a synchronized state, Ref. [91] requires a function h(gx , gy ) with either h = 0 or h → 0 asymptotically, with the choices of gx , gy and h determining the type of synchronization. This approach leads to the idea that there are different kinds of synchronization which might be captured in a single formalism. In fact, Ref. [92] demonstrated that this formalism may be simplified and generalized, in such a way that it captures all the cases that the above approach does along with an entire class that were missed. Following the reasoning of Ref. [92], let us assume that a system Z ∈ R m1 +m2 is divided into two subsystems, X ∈ R m1 and Y ∈ R m2 .

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Typically, when one thinks to synchronized states, one associates to them the fundamental concept of predictability of one system’s states from measures of the other system’s variables, that is one means that an event in one subsystem (say y) corresponds always to a particular event in the other subsystem (say x). In a more mathematical language, these events can be identified with points in the phase or state space and one can capture the notion of predictability by saying that there exists a function from X to Y such that a particular point in X is mapped, uniquely, to one point in Y. However, when one looks for evidence of synchronization in experiments or in numerics, one realistically never has data falling right on a given x˜ or on a given y. ˜ Rather, the normal situation is that the closer x(t) is to x˜ the closer y(t) is to y. ˜ This latter statement is rigorously attained with a continuous function, that is a mapping of the trajectories of x(t) close to x˜ near to y˜ by a function that is continuous at the point (x, ˜ y). ˜ To extend this approach, a more stage is necessary. Indeed, when one considers a curved one-dimensional manifold in a two-dimensional phase space (m1 = m2 = 1), the general case does not allow to construct a continuous function from x to y. But, if one further assumes a transformation F :Z → W (w = F(z)) that transforms the original curved manifold into a straight line, now the two projections of the transformed system u = P1 (w) ∈ U, v = P2 (w) ∈ V would form a synchronization manifold and one could define a synchronous state at any pair of points (u, ˜ v). ˜ From this example we learn that, in order to investigate synchronization states, the separation of the global system into subspaces is a very delicate and crucial operation, and should be performed only after a proper transformation of the global phase space is selected to disentangle “hidden” or latent synchronization features. Another important point to remark is that, for some applications, the dimensions of the final space may differ from the ones of the original phase space. Therefore, for the sake of generality, in the following we will refer to a function F : R m1 +m2 → R d1 +d2 , where d1 + d2 do not necessarily add to m1 + m2 = m. At this point, one has all the main ingredients that are needed to build up a general definition of synchronization: • (1) a function from the original phase space z to a new phase space (u, v), • (2) the projections P1 and P2 onto the components of the new space, • (3) the synchronization relation, a continuous function. A first step is refining the definition of continuous function to include consistency with the dynamics.

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D EFINITION . A function f is a synchronization function at (u, ˜ v) ˜ if (a) v˜ = f (u); ˜ (b) it is continuous at u˜ and (c) it is consistent with the dynamics (u(t), v(t)) locally, that is, if δ and are a valid pair for the continuity property (|u − u| ˜ < δ implies |f (u) − v| ˜ < ), then the dynamics is such that if |u(t) − u| ˜ < δ we have |v(t) − v| ˜ < . The above definition is tantamount to say that, near (u, ˜ v) ˜ the function f well describes the predictability of subsystem V from subsystem U. After this preliminary discussion, one needs to pay attention to more formal aspects, as initial conditions and time. D EFINITION . Let B be the basin of attraction of an attractor A for a dynamical system Z ⊂ R m , and let P1 and P2 be projectors from R d1 +d2 to R d1 and to R d2 , respectively. For a given function F : R m → R d1 +d2 , a dynamical system Z ⊂ R m contains locally synchronous subsystems in z˜ ∈ A if ∀z0 ∈ B there is a time T such that for t T a synchronization function exists at u˜ = P1 F(˜z) , v˜ = P2 F(˜z) . A first, important remark is that this definition is local, i.e., one can think of the subsystems as having properties u and v which are synchronous only near part of the trajectory (close to z˜ ), but cannot say what the relationship is between u(t) and v(t) anywhere else in the attractor. The following step is obviously that of extending the above definition to the entire trajectory, so as to have a single continuous function on the whole image of the attractor under F. This extension, in spite of being just a matter of having enough points of local synchronization, is a very delicate process. Indeed, one wants every point on the trajectory be mapped by a unique continuous function between the two subsystems u and v, while the above definition warrants that a given function is associated with each synchronization pair (u, ˜ v), ˜ and the functions may be different in their local continuity, that is the valid δ and pairs do not need to have any particular relationship between different synchronization functions. In order to cope with this situation, and to provide a rigorous extension in a theorematic way, Ref. [92] added a further feature, namely a “covering” property, and proved that a single function which maps the U -projection of F (A) (P1 (F(A)) = U) to the V -projection of F (A) (P2 (F(A)) = V) can be attained by means of a set of synchronization points on the attractor which provide a “covering” property. D EFINITION . If {ui } is a set of points on U and {fi } is a set of continuous functions, one associated with each ui , from U to V, then the functions provide a

3.10. A mathematical definition of synchronization

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continuity covering of U if ∀ε > 0 the set of all valid δi ’s associated with ε (one for each (ui , fi ) pair) covers the set U. With this in mind, Ref. [92] was able to demonstrate the following theorem: T HEOREM . If the subsystem U contains a set of synchronization points {ui } and the associated functions {fi } provide a continuity covering of U, then there exists a unique, global, continuous synchronization function f : U → V. P ROOF. Let us first demonstrate that the function f is uniquely defined. To this purpose, one proceeds by absurdity, and supposes there exist two different realizations of the dynamics z1 ∈ A and z2 ∈ A such that P1 F(z1 ) = P1 F(z2 ) = u, P2 F(z1 ) = v1 , P2 F(z2 ) = v2 . By calling η = |v1 − v2 | the distance between the two images of z1 and z2 in R d2 , one then picks ε < η/2. Furthermore, let us call uk the synchronization point whose neighborhood of radius δk (ε) contains u (the covering property warrants its existence), and let us call fk the associated synchronization function. Since fk is (by definition) consistent with the dynamics, one has fk (uk ) − v1 < ε and fk (uk ) − v2 < ε. These two inequalities may be added, and, by use of the triangular inequality, one obtains |v1 − v2 | < η, which contradicts the hypothesis. As a consequence, a unique function f exists mapping all points u ∈ U into the corresponding points v = f (u) ∈ V. The second part of the proof is to show that f is continuous at all points u ∈ U.


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By definition, ∀ε/2 there is a δj associated with one of the synchronization points uj such that |u − uj | < δj . Let us pick δ > 0 so that the set Sδ = u : |u − u| < δ is completely contained in the set of points within δj around uj . Due again to consistency of fj with the dynamics, ∀u ∈ Sδ one must have f (u ) − fj (uj ) < ε/2. On the other side, one also has fj (uj ) − f (u) < ε/2. Therefore, by use again of the triangular inequality, it comes out that f (u ) − f (u) < ε whenever |u − u| < δ, quod erat demonstrandum!

The above formalism provides a definition of perfect synchronization. In many natural and experimental systems, however, noise or finite measurement resolutions are unavoidable. It is, therefore, mandatory to introduce a fuzziness parameter, setting up a minimal coarsening scale at which states in one projected set may be associated with states in the other. Ref. [92] solved this issue, by providing the following definition of σ -synchronization: D EFINITION . For a given function F : R m → R d1 +d2 , a dynamical system Z contains locally σ -synchronous subsystems in z˜ ∈ A if ∀z0 ∈ B there is a time T such that ∀ε > σ ∃δ > 0 such that t T and P1 F Φ(t, z0 ) − P1 F(˜z) < δ ⇒ P2 F Φ(t, z0 ) − P2 F(˜z) < ε. It is evident that this last definition recovers the previous one for σ → 0. On its turn, σ = 0 is tantamount to saying that the consistency of the synchronization function with the dynamics holds only up to a minimum scale, which gives the minimal coarsening or precision scale that must be used in the experiment to detect synchronization features. An important remark is the following: even though the value of the fuzziness parameter is not constrained in the definition, if σ is larger than the diameter of P2 (F(A)), then the above is trivially satisfied ∀δ > 0 and ∀z0 ∈ B. The only

3.11. Synchronization of structurally nonequivalent systems

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relevant cases are the ones in which σ is considerably smaller than the diameter of P2 (F(A)). Finally, global σ -synchronization may be identified as the situation where local σ -synchronization is displayed regardless on the particular choice of z˜ ∈ A. With the help of the above set of definitions, Ref. [92] pointed out that different synchronization phenomena can be encompassed within such a single framework.

3.11. Synchronization of structurally nonequivalent systems Up to now, we have described the situation of a coupling involving either identical dynamical units, or nonequivalent systems, i.e., systems where the difference is limited to a rather small parameter mismatch, that does not substantially affect the main structural properties of the attractors where the subsystems’ trajectories are developing. It happens frequently in nature, however, that the coupled systems are structurally nonequivalent. This means that the two attractors where the subsystems’ trajectories are developing might well have different dimensionality, or different structural properties. Therefore, it is essential to extend the concept of synchronization to such structurally nonequivalent systems. Historically, this effort begun with a study of synchronization phenomena for large parameter mismatches. Particularly, Ref. [93] addressed the study of the appearance of synchronized collective motion in systems with more than 50% mismatch in parameters. Along the present and the following section, we will focus on two different situations, namely (i) the synchronized motion emerging as a consequence of the coupling of systems having high and different fractal dimensions [94], and (ii) the case of coupled chaotic systems with multiple coexisting attractors [95]. Let us then start illustrating the example of a symmetric coupling between two chaotic systems such that the first (when uncoupled) would give rise to a solution x1 embedded within a chaotic attractor having fractal dimension D1 , while the second (when uncoupled) would produce a dynamics x2 onto a chaotic attractor with fractal dimension D2 (with D1 significantly different than D2 ). Now, suppose that the two systems would evolve close to a complete synchronization regime for some value of the coupling strength, for which x1 (t) x2 (t) x(t). It is then natural to ask which is the fractal dimension associated with the synchronization motion x(t). Ref. [94] has addressed this issue, and has shown that synchronized states can arise for structurally nonequivalent systems either in a chaotic manifold, with dimension lower with respect to both uncoupled systems, or even in a periodic manifold. Following Ref. [94], we illustrate the issue by specializing the example made above to the case of two coupled delayed dynamical systems (DDS).

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The specific example of coupled DDS we are referring to is represented by a pair of symmetrically coupled Mackey–Glass systems, whose equations read: x1,2 (t − T1,2 ) 1 + x1,2 (t − T1,2 )10 + ε x2,1 (t) − x1,2 (t) .

x˙1,2 = −0.1x1,2 (t) + 0.2

(3.36)

In equation (3.36), x1,2 are real variables, T1,2 are delay times, and 0 ε < 1 is a coupling strength. Notice that, as soon as T1 = T2 , we have two identical DDS, and the synchronization scenario (fully studied in Refs. [96,97] even in the high dimensional chaotic case) is very similar to that of coupled identical systems. Let us now, instead, focus our attention to the case of nonidentical DDS (T1 = T2 ). Now, it is very well known that the fractal dimension of a DDS depends linearly upon the delay time [98,99]. This means that, if one selects T1 very different from T2 and chooses both delay times T1,2 to be sufficiently large, then at ε = 0 the two systems (3.36) generate high dimensional chaotic signals with rather different fractal dimensions, and therefore they come out to be confined within structurally different chaotic attractors. In this context it is interesting to investigate the effect of ε = 0 in equations (3.36). Let us then select T1 = 100 and T2 = 90. The two signals x1 (t) and x2 (t) at ε = 0 are shown in Figure 3.29(a), and they appear to be uncorrelated (Figure 3.29(b)). A first gradual increase in the coupling strength ε has the effect of building up correlations between x1 and x2 , consistently with what observed in Ref. [6] for a symmetric coupling between a chaotic and a hyperchaotic Rössler system. Eventually, the coupled system of equations (3.36) reaches a synchronized chaotic behavior (Figure 3.29(c), ε = 0.3). A further increase in the coupling strength leads to a sharp transition toward a periodic state, which is reached for large ε values (Figure 3.29(e), ε = 0.65), where the coupled system of equations (3.36) collapsed onto a simple periodic attractor. As a result, a large structural change in system (3.36) is associated with the increasing of ε. The quantitative characterization for the arousal of synchronization in system (3.36) can be obtained by making use of the indicator for generalized synchronization of equation (3.34). In Figure 3.30(a) r is reported as a function of m2 , at fixed m1 = 25 for ε 0.1, as well as for m1 = 15 at ε > 0.1. On the other hand, Figure 3.30(b) reports r as a function of ε for m2 = 35, and m1 = 25 (ε 0.1), m1 = 15 (ε > 0.1). One can easily see the primer of a synchronized state at ε 0.15.


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Figure 3.29. (a,c,e) Time evolution of signals x1 and x2 for ε = 0 ((a): uncoupled case), ε = 0.3 ((c): synchronized chaotic state), and ε = 0.65 ((e): synchronized periodic state). (b,d,f) Projections of the attractor of the system (3.36) on the plane (x1 , x2 ) for ε = 0 (b), ε = 0.3 (d), and ε = 0.65 (e). Notice that the gradual increase in the coupling strength ε has the effect of building up correlations between x1 and x2 , up to the point at which the coupled system of equations (3.36) collapses onto a simple periodic attractor. Reprinted with permission from Ref. [94]. © 2000 The American Physical Society

3.11.1. From chaotic to periodic synchronized states In the case of structurally equivalent systems the emergence of synchronization is often associated with a continuous smooth variation in the Lyapunov spectrum, eventually leading to sequential changes in the signs of the positive Lyapunov exponents.

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Figure 3.30. (a) The mutual false nearest neighbor parameter r [9,87] as a function of the embedding dimension m2 for ε = 0 (upper triangles), ε = 0.05 (circles), ε = 0.1 (lower triangles), ε = 0.2 (squares) and ε = 0.5 (diamonds). Other parameters are m1 = 25 for ε 0.1, and m1 = 15 for ε > 0.1. (b) The same MFNN parameter as a function of ε for a fixed m2 = 35. The other parameters are as above. Reprinted with permission from Ref. [94]. © 2000 The American Physical Society

For the case considered here, instead, the synchronized motion arises initially on top of a chaotic dynamics, but, at larger coupling strengths, the occurrence of a periodic synchronized state is associated with an abrupt transition in the Lyapunov spectrum, where many positive Lyapunov exponents passes to negative values at once. This is manifested clearly in Figure 3.31, that reports the Kaplan–Yorke or Lyapunov dimension of equation (3.36) (Figure 3.31(a)) as well as the number of


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Figure 3.31. (a) Kaplan–Yorke dimension of equation (3.36) as a function of the coupling strength ε. (b) Number of positive exponents in the Lyapunov spectrum vs. ε. (c) The ten largest exponents in the Lyapunov spectrum vs. ε. In all cases the calculations have been performed over a time t¯ = 1,000,000, corresponding to 10,000 delay units of the system with larger delay. The structural collapse at ε 0.6 is marked by a sharp transition in the Kaplan–Yorke dimension (a) and by the fact that many positive Lyapunov exponents goes to negative value at once (c). Reprinted with permission from Ref. [94]. © 2000 The American Physical Society

positive Lyapunov exponents (Figure 3.31(b)) vs. the coupling strength. One can notice the initial (at small ε) slow continuous decreasing process of the Lyapunov dimension, in a way consistent with what happens for structurally equivalent systems.

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At larger couplings, however, two different dynamical regimes arise. This first corresponds to the appearance of generalized synchronization (0.15 < ε), where a plateau in the Lyapunov dimension around D 7.2 − 7.5 sets in for 0.15 < ε < 0.6. This indicates that generalized synchronization is initially realized over a high dimensional chaotic state. Correspondingly, the number of positive Lyapunov exponents does not change in the range 0.15 < ε < 0.6. Notice, furthermore, that the dimension of the generalized synchronization manifold comes out to be smaller than both initial dimensions of the uncoupled systems, indicating that the process of adjustment of trajectories selects a third attractor whose dimensionality is smaller than that of the initial uncoupled attractors. A second regime is encountered for 0.6 < ε. Here, a sharp transition in the Lyapunov dimension is found, leading to the stabilization of a final periodic state. It is important to remark that such a transition corresponds to a sudden change in the Lyapunov spectrum, wherein all residual positive Lyapunov exponents suddenly jump to negative values at once (Figure 3.31(b)). This is confirmed by the calculation of the ten largest Lyapunov exponents in the spectrum of equations (3.36) as functions of the coupling parameter ε (Figure 3.31(c)) over long time series, involving several thousand of delay units. The main conclusion of Ref. [94] is that generalized synchronization can be set in a pair of high dimensional structurally nonequivalent systems. In this case, the collective motion leads to an adjustment of the two systems into a manifold whose dimension can be smaller than any one of the dimensions of the uncoupled systems.

3.12. Systems with coexisting attractors Another important class of systems where synchronization can take place is that of multistable chaotic systems, i.e., chaotic systems where the coexistence of several chaotic attractors occurs for a given set of parameters. This implies that, depending on the initial conditions that are set, the system can visit one or another of such coexisting attractors, and the basin of attractions (the sets of initial conditions that evolve into the specific coexisting attractors) may be interwoven in a very complicated way. The importance of studying synchronization phenomena emerging in these systems is due to the fact that multistability was indeed observed in different fields of science, including electronics [100], lasers [101], mechanics [102], biology [103], nuclear physics [104], chemistry [105], and economy [106]. A relevant issue is the understanding of the synchronization scenario arising when the state of a multistable system is coupled with another identical system being in another state.


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The answer to this question manifests inherent difficulties, when considering, e.g., two unidirectionally coupled chaotic bistable systems in a master–slave configuration. Intuitively, one could think that, as the coupling strength increases, the slave system will gradually accommodate its state to that of the master system and then the problem would reduce to the well-known case of two identical chaotic monostable systems. However, this naive representation is only partially correct. It has been, indeed, demonstrated [95] that the synchronization scenario of coupled multistable systems is much richer and more complicated, including different types of synchronization, intermittency, shift of the natural oscillator frequency, and frequency locking. Following Ref. [95], let us illustrate the case with the example of two identical unidirectionally coupled piecewise linear Rössler-like electronic circuits [107– 109], described by: dx1 dτ dx2 dτ dy1 dτ dz1 dτ where

= −αx1 − z1 − βy1 ,

= −αx2 − z2 − β y2 + ε(y1 − y2 ) ,

dy2 = x2 + γ y2 + ε(y1 − y2 ) , dτ dz2 = g(x1 ) − z1 , = g(x2 ) − z2 , dτ = x1 + γ y 1 ,

g(x1 ,2 ) =

0, μ(x1,2 − 3),

(3.37) (3.38) (3.39)

if x1,2 3, if x1,2 3,

is a piecewise linear function, τ = t × 104 s (t being the real time), α = 0.05, β = 0.5, γ = 0.3, μ = 15, and ε ∈ [0, 1] is the coupling strength. Though being identical, if the master and slave systems evolve from different initial conditions at zero coupling, their trajectories will be confined in different chaotic attractors with natural frequencies fm and fs . After having prepared the systems in these conditions, Ref. [95] described the effects of gradually increasing the coupling strength. Initially, one does not see any appreciable effect on dynamics of the slave oscillator up to ε = 0.005. The two systems evolve in a fully unsynchronized manner, and the phase difference between the slave and master oscillations increases linearly with time. At a certain critical value of the coupling strength (εc = ε1 = 0.00505), the slave system starts to switch intermittently between its two coexisting chaotic attractors (Figure 3.32(a)).


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Figure 3.32. The dynamics of slave system at ε = 0.0051. (a) Time series of the master and slave systems demonstrating intermittent switches between coexisting attractors. (b) The temporal behavior of the phase difference. Notice that, while φ increases linearly in time as far as the systems stay in different attractors, it instead fluctuates around a certain value as long as the systems stay in the same attractor. (c) The random walk process suffered in time by the phase difference inside the windows where the systems stay in the same attractor. (d) The probability distribution of the phase difference inside the windows. The line is the Gaussian fit. A negative phase difference means anticipated synchronization. Reprinted with permission from Ref. [95]. © 2006 The American Physical Society

Now, the phases of the master and slave oscillators can be defined as φ(t) = 2πk + 2π

t − tk tk+1 − tk

(tk being the time of kth maximum of the corresponding signal), accordingly to the definition of phases through the associated Poincaré section.


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Within the windows where the slave and master oscillators stay in the same attractor the average time difference between successive peaks of the master and slave oscillations is equal to the natural period of the chaotic oscillations, and one can approximate m s tk+1 − tkm ≈ tk+1 − tks ≈ 1/fm .

Therefore, the phase difference between the peaks of the slave and master oscillations with the same number k can be written as φ = 2π tks − tkm fm (where tks and tkm are the times of kth maximum of the slave and master oscillations). As it can be seen in Figure 3.32(b) and (c) φ within these windows fluctuates randomly in the range δφ ≡ φ max − φ min ≈ 2π (φ max and φ min being the maximum and minimum phase difference) featuring a normal probability distribution (Figure 3.32(d)) and a most probable phase difference φ ≈ −π, which is the signature that the preferred phase synchronized state occurs in the anti-phase regime. Intuitively, this regime can be understood as follows: the chaotic properties of the master system act as a noise term on the slave system for this so weak coupling. Indeed, as ε increases, the jumps to the synchronization regime occur more and more frequently and δφ decreases (Figure 3.33). This means that the phases inside the windows become synchronized within a certain range of the phase difference. This regime has been referred to as intermittent phase synchronization [95]. The average anticipation time decreases with increasing ε (φ → 0) while δφ → 0. The mean duration of the windows (laminar phase), tL also decreases and eventually at ε ≈ 0.1 the windows of intermittent phase synchronization disappear. A nontrivial result is that the oscillations of the slave system anticipate the oscillations of the master system and the maximum anticipation time is of the average period of the chaotic oscillations, 1/fm (Figure 3.33). This means that the slave system is synchronized not with the present state of the master system but with its future state. In order to understand the transition route from asynchronous motion to perfect phase synchronization inside the intermittent phase synchronization windows, Ref. [95] derived the scaling relationships between tL and ε.


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Figure 3.33. The average phase difference (closed dots) and fluctuation range (open dots) versus the coupling strength inside windows of intermittent phase synchronization. Notice that, as ε is increased, both the anticipation time and δφ decrease. Reprinted with permission from Ref. [95]. © 2006 The American Physical Society

tL has nonmonotonic dependence on ε, and it is possible to distinguish three distinct regions. Precisely, with increasing ε, first decreases (0.0051 < ε < 0.01), then it increases (0.01 < ε < 0.028), and finally it decreases again (0.028 < ε < 0.1). It seems that the system has two saddle-node bifurcation points corresponding to the maximum duration of the IPS windows at ε1 and ε2 . In the first and last regions, the mean duration of laminar phase obeys the power law tL ∼ (ε − εc )p

(3.40)

with two different scaling exponents p = −1/2 (Figure 3.34(a)) and p = −1 (Figure 3.34(b)). The first exponent corresponds to the region near the onset of intermittency in the vicinity of the critical point ε1 and the second one near εc = ε2 = 0.028. The first scaling exponent is a characteristic of type-I intermittency. The critical exponent of −1 is a signature of on-off intermittency. In the middle range of coupling, 0.01 < ε < 0.028, the scaling exponent is positive, i.e., increases with increasing ε.


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Figure 3.34. Power law dependencies of the mean duration of intermittent phase synchronization windows vs. the coupling strength. The straight lines have slopes of (a) −1/2 and (b) −1 that characterize type-I and on-off intermittencies, respectively. Reprinted with permission from Ref. [95]. © 2006 The American Physical Society

Thus, the evolution of the bistable chaotic system from asynchronous behavior to perfect phase synchronization is realized through type-I and on-off intermittencies. Besides IPS and anticipated synchronization another interesting phenomenon can be observed in the coupled bistable chaotic systems. At relatively strong coupling (ε > 0.25) the fundamental frequency of the chaotic attractor of the slave system, fs , begins to decrease moving towards the period-doubling frequency of the master system, fm /2. As a result, slips of

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phase-synchronized period-doubling oscillations appear in the time series. For this reason, this feature has been termed period-doubling phase synchronization. With a further increase in ε, fs approaches fm /2 and the windows of period doubling phase synchronization becomes larger. Finally, fs becomes completely locked by fm /2 (at ε > 0.5) and a stable period doubling phase synchronization regime is observed (see Ref. [95] for all details on this regime).

3.13. Methods and tools for detecting synchronized states After having described the main synchronization phenomena that take place in coupled identical or nonidentical systems, it is important to focus on the main tools that are used to identify and characterize these phenomena in the analysis of data coming from experiments or measurements of natural processes. These tools, indeed, are essential to discover interdependence, or functional relationships, between independent measures of the dynamics produced by coupled subsystems, and provide the most important instruments to detect and model synchronization phenomena occurring in laboratory experiments and in nature. At the beginning of this discussion, it has to be highlighted that most of laboratory or natural systems often display the following features: (i) they are almost never fully identical and in most cases one has to consider that the recorded dynamics is nonstationary, i.e., it is affected by considerable variations or fluctuations of the system parameters; (ii) almost all measurements coming from natural processes are unavoidably affected by noise, which can be a simple measurement noise due to the instrumentation, or it may be a nontrivial noise process that inherently affects the dynamics of the two subsystems; (iii) in many cases, the model description of the observed phenomena is very complicated, in many other cases it is even unknown, as simple and tractable models are often not available. Most likely the only accessible information is the recorded time series from different subsystems. This means that effective tools and methods should be model independent, and possibly data driven, to avoid the inherent uncertainty in the knowledge of the underlying dynamical models; (iv) while in our discussions on model systems, the system parameters were conveniently tuned to discover the different synchronization processes and to study the transitional scaling properties occurring from one to another of the synchronous states, in most natural systems, there is no access to such system parameter nor one can perform a control of them. This implies the intrinsic difficulty of identifying or characterizing states (as, e.g., phase synchronization) that are intrinsically related to the maintenance of synchronization over a region in parameter space.

3.13. Methods and tools for detecting synchronized states

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Taking into account these important points, we now summarize the main tools that are available for the description of synchronization in laboratory and natural systems. It is easy to understand that, as long as one is interested in detecting complete synchronization phenomena, one can simply monitor the difference between the measured time series, and look for a vanishing of such a difference over a relatively long time scale. However, as far as other synchronization phenomena are concerned, the identification of complicated or even weaker interdependence is not straightforward and requires more sophisticated approaches. It has to be mentioned that recently, different approaches have been put forward, that made use of the theory of recurrences and recurrence plots allowing for the study and quantification of the interaction between systems and providing tools for the identification and analysis of the arousal of synchronization phenomena. As for these latter techniques, the reader can find a very nice and complete overview, and an exhaustive associated literature in Ref. [110]. In the following, instead, we will concentrate on more classical techniques for detection of synchronized and collective behaviors in coupled systems. 3.13.1. Detection of a functional relationship A first (very well known and commonly used) method to search interdependence between two time series {xi } and {yi } sampled with a time step t is to calculate the cross-correlation function C(X, Y, τ ) = xi − x yi−l − y , (3.41) where . . . stands for average over time, and τ is a variable time lag. In the majority of cases, however, the simple inspection of linear relationships (it is important to emphasize that C(X, Y, τ ) retains information only on the linear interdependence between the time series) does not furnish sufficient information to support a claim for the existence of underlying synchronization processes, and one has to make use of tools that detect and quantify nonlinear interactions. For this purpose, it is possible to make use of concepts from information theory, such as the so called mutual information I (X, Y ) = H (X) + H (Y ) − H (X, Y )

(3.42)

which is based on the Shannon entropies H (X) and H (Y ) of both time series and the entropy H (X, Y ) of the joint probability function [111]. One should be warned, however, that a proper and reliable estimate of the stationary probability distributions p(X), p(Y ) and p(X, Y ) requires in general very long and stationary data sets, which, as we discussed, are hard to find in real measurements, and therefore, the application of this specific tool may be strongly hampered in several cases.

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3.13.2. Embedding and multivariate data A variety of techniques for the analysis of multivariate time series are based on the assumption that a deterministic dynamics is responsible for the recorded complex time series [87,112], and that one would be able to properly reconstruct an equivalent phase space in a high enough dimensional embedding space. The phase space structure of the dynamics then should be unfolded and reconstructed equivalently in such an embedding space with the use of the information coming from only a scalar time series according to Taken’s embedding theorem. In order to construct proper vectors in a generic m dimensional embedding space, one makes use of time-delay (which is generally a multiple of the sampling time, e.g., τ = lt) coordinates of the time series as xi = (xi , xi−l , . . . , xi−(m−1)l ). For multivariate time series, the embedding parameters m and l can be different and often should be chosen individually for X and Y . The preliminary fundamental task to reconstruct correctly the strange chaotic attractors from observed scalar data is in general a very delicate issue, as a correct determination of the system dimensionality is an essential problem to be solved in order to approach correctly any following steps of a nonlinear data analysis. A popular method that is used for measuring the minimal embedding dimension is the so-called false nearest neighbor (FNN) method, originally introduced by Kennel et al. [113], and later improved and reelaborated in order to face specific analysis tasks [114]. The method consists in marking as false nearest neighbors at dimension m those pairs of nearest neighbors m-dimensional embedded vectors whose distance at dimension m + 1 exceeds a given number of times their distance at dimension m, thus accounting for possible self-intersections of the flow due to insufficient dimensionality in the embedded space. A vanishing fraction of FNN marks the minimum dimensionality needed to properly reconstruct the chaotic flow. This technique has been later improved [114], also complementing this analysis with the one on the signal surrogates [115]. Ref. [116] has recently extended a technique for solving the problem of the dimension reconstruction to the case of multivariate data analysis, that is to the case in which an observer is presented with a system composed by n weakly coupled nonidentical dynamical subsystems (of dimensions l1 , l2 , . . . , ln , respectively), and extracts separately scalar quantities xi (t) out of each subsystem i (i = 1, . . . , n). If the observer is interested in probing global properties of the system under study (and if the subsystem variables are all to all coupled), then the usual reconstruction methods work regardless on the particular variable xi (t) on which embedding is performed.


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However, this gives information on the full phase space dimensionality. There are relevant cases, for which the determination of the dimensionality of each subsystem is needed. The problem can be stated as follows. Consider having n weakly coupled nonidentical dynamical systems, and suppose that an observer is able to measure separately scalar quantities xi (t) out of each subsystem i (i = 1, . . . , n). In order to properly reconstruct the dimensions l1 , l2 , . . . , ln , let consider the vector z ≡ x1 (t), x1 (t − τ1 ), x1 (t − 2τ1 ), . . . , x1 t − (m1 − 1)τ1 , x2 (t), x2 (t − τ2 ), . . . , x2 t − (m2 − 1)τ2 , . . . , xn (t), xn (t − τn ), . . . , xn t − (mn − 1)τn , (3.43) where τi (i = 1, . . . , n) are n different embedding times. z ∈ Rm (m = i=1,n mi ) is a vector whose first (second, third, . . . , nth) m1 (m2 , m3 , . . . , mn ) components result from the embedding of the x1 (x2 , x3 , . . . , xn ) scalar variable with embedding time τ1 (τ2 , τ3 , . . . , τn ). The embedding times τi can be different from each other, since different observed variables xi (t) may show different mutual information properties. Suppose now to be at dimension m realized with a choice of an initial set of subspace dimensions {mi } (usually one begins with mi = 1, i = 1, . . . , n), and to consider all m-dimensional vectors zj , j = 1, . . . , N , N being the total number of available measurements. Ref. [116] associated to each vector zj its nearest neighbor zN N,j at dimension m, and introduced n counters Ni (m) (i = 1, . . . , n), and a given threshold σ . For each pair of nearest neighbors zj , zN N,j the distance

|zj,l − zN N,j,l | d(zj , zN N,j , m) = l=1,...,m

can be evaluated. When now one wants to pass from dimension m to dimension m + 1, one immediately realizes that such an operation can be performed in n different ways. Precisely, from m ≡ (m1 , . . . , mi , . . . , mn ) one can pass to any space m + 1 ≡ (m1 , . . . , mi + 1, . . . mn ) In those spaces, the new distances are di (zj , zN N,j , m + 1).

(i = 1, . . . , n).


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The condition di (zj , zN N,j , m + 1) σ d(zj , zN N,j , m) can be taken as a signature of the falseness of nearest neighbors with respect to increasing by one the dimension of the ith subspace. Consequently, the counter Ni (m) is increased by one. After having probed all pairs of nearest neighbors at dimension m, the set of counters Ni (m) (i = 1, . . . , n) retain information on how many nearest neighbors are false with respect to increasing by one the dimension of the corresponding ith subspace. For any Ni (m) above a preassigned threshold δ one then increases by one the dimension of the corresponding subspace, and performs the whole process again at dimension m + p, p being the number of Ni (m) counters that overcome δ. The process can be stopped when all Ni (m) are below δ at once, thus gathering simultaneous information on both the dimension of the full reconstructed phase space, and the dimensions mi of each subsystem. Once the embedding space has been reconstructed properly, many methods exist to reduce the measurement noise and to compute invariant quantities, such as Lyapunov exponents λi and or the information dimension Di . These quantities allow to make some initial prediction of the time series, and also allow to construct local or global models for the observed dynamics. 3.13.3. Interdependence between signals According to the definition of generalized synchronization, the two (sub)systems are connected by a functional relationship Y = ψ(X), i.e., whenever two states xi and xj are close to each other in the first subspace, the corresponding states yi and yj in the second subspace are also close to each other. Based on this concept, Ref. [9] proposed the method of mutual false nearest neighbor to detect generalized synchronization. Actually, such a method represents a variation of a somewhat more general method proposed in Ref. [117], which considers closeness not only by the nearest neighbor point, but to an region in the neighborhood. Ref. [117] introduces the mean conditional dispersion

1/2 2 σXY () = yi − yj Θ − xi − xj / Θ − xi − xj i =j

i =j

(3.44)

where Θ it the Heaviside function with Θ(z > 0) = 1 and Θ(z 0) = 0. When X and Y are in generalized synchronization, σXY () should decrease with decreasing , while if they are not related, σXY () is independent of .


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With this measure, interdependence may be detected while linear correlation indicates no significant interrelation. With a similar idea, Ref. [118] developed a set of statistics for mathematical properties, such as continuity and differentiability, of maps between time series in terms of probabilities or confidence levels. In the similar spirit, other variations of the method have been proposed in Ref. [119] and investigated further in Refs. [120,121]. Here the difference is to consider k nearest neighbors ri (j ) (j = 1, . . . , k) of a state xi and k nearest neighbors si (j ) (j = 1, . . . , k) of the state yi . The squared mean distance and conditional distance then are given by 1

(xi − xri (j ) )2 , k k

Ri(k) (X) =

j =1

1

(xi − xsi (j ) )2 , k k

(k)

Ri (X|Y ) =

j =1

respectively. With these quantities, a measure for interdependence can be defined as S (k) (X|Y ) =

N (k) 1 Ri (X) , (k) N i=1 Ri (X|Y )

(3.45)

which takes a value close to 0 when X and Y are independent and a value close to 1 when generalized synchronization occurs. Based on the conditional distance, another measure was also proposed similar to equation (3.45), but using the distance of random points

Ri = (N − 1)−1 (xi − xj ) j =i

instead of that from the k nearest neighbors for comparison, which reads H (k) (X|Y ) =

N 1 Ri (X) . (k) N i=1 Ri (X|Y )

(3.46)

Both measures have proven to be quite useful in real data applications [119] and simple toy models [120,121]. 3.13.4. Predictability of time series A few other methods exist that are based on predictability of time series. In particular, Ref. [122] used mutual prediction for detecting dynamical interdependence. With the k nearest neighbors ri (j ) (j = 1, . . . , k) of a state xi , the average value after L steps translation can be taken as the prediction of the state L steps ahead, e.g., 1

xri (j )+L , k k

xˆ i+L (X) =

j =1

(3.47)


100

and the average prediction error is N 1 xi+L − xˆ i+L (X), e(X) = N i=1

which then is further normalized by the prediction error when taking the mean value of the time series as the predicted value, i.e., emean (X) =

N 1 xi+L − x, N i=1

giving (X) =

e(X) emean (X)

(3.48)

(X) = 1 indicates that there is no predictability in the time series. Similarly, prediction can be performed with mutual nearest neighbors si (j ), so that 1

xsi (j )+L , k k

xˆ i+L (X|Y ) =

j =1

and e(X|Y ) =

N 1 xi+L − xˆ i+L (X|Y ). N i=1

Likewise, the normalized mutual prediction error can be calculated, (X|Y ) =

e(X|Y ) . emean (X)

(3.49)

When X and Y are in generalized synchronization, bidirectional mutual prediction can be observed, i.e., (X|Y ) < 1 and (Y |X) < 1. A recently introduced novel approach is based on the interesting idea of introducing a mixed state embedding for multivariate time series [123]. The idea is based on the fact that, when two nonlinear systems are coupled, the dynamics of the whole system can be unfolded and reconstructed in a higher dimensional embedding space and predictability of one subsystem dynamics from the observation of the other subsystem dynamics can be obtained consequently with nearest neighbors in the mixed state space pi (m, n) = (xi , xi−l , . . . , xi−(m−1)l ; yi , yi−l , . . . , yi−(n−1)l ). With k nearest neighbors ri (j ), j = 1, . . . , k, in the mixed state space, the prediction value xˆ i+L (X) of L (L = 1 and k = 1 are considered in Ref. [123]) step


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ahead can be computed as in equation (3.47), and the prediction error is given by e(X, m, n) =

N 1 xi+L − xˆ i+L (X). N

(3.50)

i=1

Since the nearest neighbors in subspace X or Y may turn out to be false nearest neighbors in the mixed embedding, prediction may be improved by mixed state embedding. And by examining the pattern of e(X, m, n) in the (m, n) space, interdependence between X and Y can be detected. 3.13.5. Coupling direction A very important issue in the analysis of functional relationship is the detection of the coupling direction. Depending upon the specific coupling configuration, indeed, there can be a strong asymmetry characterizing the interdependence between X and Y . For instance, in unidirectional coupling configurations from X to Y , Y is affected by X, but X is free and is not influenced by Y . Both the correlation function and the mutual information definitions are intrinsically symmetric when exchanging X and Y . Thus both measurements cannot provide information on coupling direction. All other measures mentioned above are in general asymmetrical when exchanging X and Y , and could be useful for detecting coupling asymmetry. The measures of asymmetry degree have a nontrivial dependence on other parameters of the data, such as the number N of data point and the number k of nearest neighbors, and in some cases, the interpretation of this information is difficult [119,121]. The information-theoretic measure, coarse-grained information rates, has been introduced [124] so that interdependence as well as the direction of information flow can be detected in coupled systems. Detecting direction of coupling in interacting oscillators has also been studied using the phase dynamics [125], which would provide a more promising detection when the interdependence between amplitudes is weaker than that of phases due to weak interaction. 3.13.6. Detection of phase synchronization Since phase synchronization represents the weakest stage of synchronization, in which only the instantaneous phases are mutually related, but the corresponding amplitudes are in general not correlated, all measures based on the assumption of the existence of a mapping (or a functional relationship) between the time series are, in most cases, not adequate to detect with confidence the emergence of such a collective motion. Therefore, the correct procedure is to first extract properly the phases of the coupled oscillators, rather then reconstructing the phase spaces of the coupled

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subsystems. The first step is then to compute with accuracy the instantaneous phases from the time series. We have already discussed that such a procedure is not generic, and one has to apply different measures of the instantaneous phases, depending on the phase coherence properties of the coupled oscillators. Here, instead, we discuss the calculation of the phase from the viewpoint of filtering. For narrow-band signals x(t), e.g., a signal from a system with a proper rotation about a reference point, phase can be obtained by means of the analytic signal V (t) using the Hilbert transform, which is equivalent to filtering the signal with a filter j 1 F (t) = δ(t) + P π t (P 1t means the principal part of 1t ) whose amplitude response is unity, and whose phase response is a constant π/2 lag at all frequency. Writing V (t) = A(t)ej φ(t) = x(t) F (t),

(3.51)

where denotes convolution, the computing of the phase then becomes a problem of filtering the signal. More importantly, one can consider other choices of filter F (t). This is particularly important for systems displaying a broad distribution of time scales, for which very likely not all time scales are synchronized. It is thus interesting and important to detect synchronization between some particular time scales and the definition of phase with filtering becomes very natural. One useful choice can be a Gaussian type of filter around a characteristic frequency ν0 with a width σ , e.g.,

σ F (t) = √ exp −iν0 t + σ 2 t 2 /2 . 2π This definition of phase was discussed in Refs. [126,127]. When phases (lifted to the whole real line, i.e., not modulated to [0, 2π]) are defined and computed for individual time series, instantaneous frequencies may also be estimated by the derivative of the polynomial function fitted to the phases on an interval essentially larger than the characteristic period of oscillations. With the phase and frequency information, one is ready to detect synchronization. Due to several effects, such as borderline of phase synchronization, imperfect synchronization, noise and nonstationarity, phases are most likely not locked perfectly, but often interrupted by phase slips. Ref. [128] presents a few methods to detect phase synchronization of noisy data: (a) Analysis of phase difference. Phase locking epochs may be manifested by horizontal plateaus in the plot of difference φn,m = nφx − mφy of lifted phases φx and φy . However, due to strong noisy or chaotic fluctuations in the signal, clear plateaux may not be easily detectable, and one should analyze the distribution of the cyclic phase


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difference Φn,m = φn,m mod 2π over a moving window, where peaks indicate preferred phase differences between the two systems. The degree of phase synchronization can be measure more quantitatively. Ref. [129] proposes two measures of the n : m synchronization index. The first one based on the Shannon entropy is defined as ρnm = (Smax − S)/Smax , where S = − N k=1 pk ln pk is the entropy of the distribution of Φn,m and Smax = ln N (N is the number of bins) is the entropy of the uniform distribution. Normalized in this way, 0 ρnm 1, and larger ρnm corresponds to higher degree of phase synchronization. The second index is based on conditional probability. The intervals of two phases φx and φy , [0, n2π] and [0, m2π], are divided into N bins. Then for each bin l, one calculates γl (tj ) =

1 iφy (tj ) e Ml

for all j such that φx (tj ) belongs to bin l and Ml is the number of points in this bin. Averaging γl (tj ) over all the N bins gives the synchronization index λnm =

N 1 γl (tj ), N l=1

which assumes a value between 0 and 1. A similar quantity can also be defined based on the phase difference Φn,m , as λ˜ nm = sin2 Φn,m + cos2 Φn,m . Similar to the conditional probability, mutual information between the distributions of the phases φx and φy can be calculated as a quantitative measure of phase synchronization degree, as demonstrated in Chapter 4. Mutual information combined with a test of its statistical significance has been used to detect phase synchronization in noisy systems [130]. (b) Instantaneous frequency ratio. It is natural to calculate the instantaneous value of the frequency ratio, i.e., the time dependent value of the ration between the instantaneous frequencies. For synchronized systems, this ratio should fluctuate around a rational number. Its advantage is that there is no need to search for appropriate values of n and m. However, due to noise, nonstationarity and short interval of synchronization epochs, this one may not be able to distinguish synchronization from occasional coincidence of frequency, and this method can be used only in addition to the analysis of phase differences.

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(c) Constructing a synchrogram. This method employs the idea of phase stroboscope: when the cyclic phase of one system attains a certain fixed value θ at a moment tk , e.g., φx (tk ) = θ , the cyclic phase of the other system is plotted vs tk to construct a m order synchrogram, i.e., ψm (tk ) = φy (tk ) mod 2πm.

(3.52)

In this graphic presentation, n:m synchronization can be visualized by n distinct horizontal lines and transitions between different ratios of phase locking can be traced, which is extremely useful for application to biological data where nonstationarity is often strong. 3.13.7. Detection of local synchronization At the end of this chapter, we would like to pay attention to the problem of detecting what is called local synchronization. We have seen, indeed, that a unifying approach to the problem of synchronization [92] refers to the emergence of local functional dependencies between neighborhoods of particular phase space configurations in the projected spaces of the two coupled subsystems. In other words, not always a unique functional relationship can be established allowing for the prediction of one system’s behavior from the measurement of the other system’s states. Rather, in many circumstances, synchronization properties can be limited to a subset of the possible states visited by the evolution of the coupled systems, thence the need of a search for local functional relationships. Implementation of a search for local functional dependencies requires two separate steps: a preliminary one in which the two interacting subsystems X and Y are properly identified within the original dynamical systems Z, and their dimensionalities measured, and a second one in which the local synchronization points (x, ˜ y) ˜ are detected. The first problem was solved recently in Ref. [116] by means of a modification of the false nearest neighbors algorithm [87], allowing for a separate measurement of the dimensionalities of weakly coupled systems in the case of emergent synchronization motions. Here, we will resume the approach of Ref. [131] that addressed the second step of the search by introducing the synchronization points percentage (SPP) indicator, and showed how one can gather information on local synchronization properties emerging in coupled chaotic systems. One starts by assuming to have N data points in Z ∈ Rm1 +m2 . By means of a proper subspace reconstruction [116], one ends up with N data points in X ∈ Rm1 and N corresponding images in Y ∈ Rm2 . One then picks a specific point x˜ ∈ X and considers its image y˜ ∈ Y.


105

The first task consists in identifying proper domains and co-domains for a statistical analysis of the existence of functional dependency. For this purpose, Ref. [131] selected a pair of positive real numbers (εk , δ) (the index k being an integer), and considered the volume Uεk ⊂ X (Vδ ⊂ Y) containing all points whose m1 -distance (m2 -distance) from x˜ (y) ˜ is smaller than εk (δ). Furthermore, Ref. [131] looked at all points in X falling within Uεk , and verified the imaging condition (that is whether or not all images of the points in Uεk fall within Vδ ). If such a condition is not verified, the strategy is to choose εk+1 < εk , and to repeat the above procedure. If for all k the imaging condition is not satisfied, the task ends with the conclusion that no local functional dependency exists in the vicinity of the chosen configuration (x, ˜ y). ˜ If, instead, for a given k˜ the imaging condition is verified, the task ends with the identification of a valid pair (εk˜ , δ), over which one has to test for the existence of a continuous functional relationship. Figure 3.35 helps in understanding the schematic representation of the procedure. In the following we will denote with U ⊂ X (V ⊂ Y) the neighborhood Uεk˜ (Vδ ) surrounding x˜ (y), ˜ and assume that m < N points fall within U . By construction, the number of points falling within V will be n m, reflecting the fact that V might host also images of points not belonging to U . The probability of a single point falling within V is P (V ) ≡ n/N, and the probability that m points fall within V by pure chance is m n Pm (V ) = P (V )m = . N This latter quantity, for reasonable choices of n, m (reasonable pairs (εk˜ , δ)), is a very small number. However, one has to fix a confidence level of comparison, for assessing existence of a local continuous function between the two neighborhoods. This problem was recently addressed, and the continuity statistics method was proposed. This consists in calculating the quantity bP , defined as bP = max B(q, m; P ), q=1,...,m

(3.53)

where B(q, m; P ) is the binomial distribution, giving the probability that q m events out of m attempts are realized for a process of elementary probability P . As said above, the presence of a single data within V has probability P (V ). The quantity bP (for P = P (V )) represents then the maximum over q of the probability that, given m points, q out of them fall into V . Hence, a level of confidence for the existence of a continuous function can be estimated in terms of the


106

Figure 3.35. Sketch of the statistical continuity analysis. The upper part shows the reconstructed trajectories in the two subspaces X and Y, and the location of the points x˜ and y. ˜ The lower part zooms on the U, V neighborhoods. In the figure, for ε = εk+1 , V contains all images of the m points in U (solid circles), plus images of other points (empty squares) from outside U . For ε = εk , some points in U (empty circles) have images outside V . Reprinted with permission from Ref. [131]. © 2004 The American Physical Society

ratio Θ=

Pm (V ) . bP

(3.54)

If Θ ≈ 1 we have no trustable information about the existence of such a functional relationship, insofar as the chance probability of having our m points in V is of the same order of the maximum probability of having events in V out of m attempts. On the contrary, if Θ 1, the chance probability of having our m points in V is negligible compared to bP . Thus one concludes that the two sets U and V are the domain and co-domain respectively of a local continuous function mapping states in X close to x˜ to states in Y close to y. ˜ This answers the practical question of predicting states in Y with an error δ from measurements of states in X with uncertainty εk˜ . The technique for characterizing synchronization introduced in Ref. [131] consists then of the three following points:


107

(i) check the imaging of neighborhoods of a given configuration x˜ into neighborhoods of y; ˜ (ii) assess the degree of confidence that such an imaging process comes from the existence of a local continuous function; (iii) repeat points (i) and (ii) for all N pairs of configurations (x, ˜ y) ˜ available in the data set. This procedure allows a classification of the different dynamical states into locally synchronized and nonsynchronized ones. As a result one can introduce the synchronization points percentage (SPP) indicator, as the ratio between the total number n˜ of locally synchronized configurations and the total number of available points N . The SPP indicator furnishes relevant information in all those cases in which synchronization states emerge locally in phase space, to detect predictability properties that are limited to some subset of the dynamics. In order to illustrate the method, Ref. [131] referred to a pair of nonidentical bidirectionally coupled chaotic Rössler oscillators. Here m1 = m2 = 3, and the subspaces X and Y contain state vectors x ≡ (x1 , y1 , z1 ) and y ≡ (x2 , y2 , z2 ) whose evolution is ruled by x˙1,2 = −ω1,2 y1,2 − z1,2 + (x2,1 − x1,2 ), y˙1,2 = ω1,2 x1,2 + 0.165z1,2 ,

(3.55)

z˙ 1,2 = 0.2 + z1,2 (x1,2 − 10). In equations (3.55), ω1,2 = ω0 ± represent the natural frequencies of the two chaotic oscillators, ω0 = 0.97, = 0.02 is the frequency mismatch and > 0 rules the coupling strength. As increases, a sequence of emerging synchronization features is observed in equations (3.55). Precisely, for < 0.036 no global synchronization (NS) is established. For 0.036 0.11 a phase synchronized (PS) regime emerges. At larger coupling strengths ( 0.145), lag synchronization (LS) is established. Eventually, the system sets in a regime which is almost indistinguishable from complete synchronization (CS). Ref. [131] applied the local detection of synchronization with a threshold value of Θ = 0.1 for the discrimination of whether or not the coupled systems display local functional relationships, performed long time simulations of equations (3.55) at several coupling strength values, and collected data points from the two scalar outputs x1 and x2 . For each , data points were collected over a time corresponding to 1.7 × 105 Rössler cycles, with a sampling frequency of 10 points per cycle. Furthermore, the standard embedding technique [132] was used to reconstruct the three dimensional vector states x and y from time-delayed coordinates of the scalar variables

108


Figure 3.36. The synchronization point percentage (SPP) indicator (see text for definition) as a function of the coupling strength . The vertical dashed lines indicate the transition points between the different synchronization regimes. The inset shows a zoom limited to the range 0.05 < < 0.15, where the phase to intermittent lag synchronization transition point is located at c 0.10. Notice the two different slopes in the linear growth of SPP for < c and > c . Notice furthermore that already during the phase synchronization regime there is an increasing percentage of local synchronization points. Reprinted with permission from Ref. [131]. © 2004 The American Physical Society

x1 and x2 , and calculation of the SPP indicator was performed on the reconstructed spaces. Figure 3.36 reports the behavior of the SPP indicator vs. the coupling strength , calculated by fixing δ so as n = 150 points are falling within V . As one expects, SPP increases monotonically as the coupling strength increases, saturating to 1 when approaching the CS regime. Interesting novel information can be extracted by inspection of SPP within those synchronization regimes, such as PS and ILS that do not correspond to global synchronization features. In particular, it is found that SPP is linearly increasing with in both regimes, but with two different slopes (see the inset of Figure 3.36). The linear increase of the indicator already within PS gives an interesting novel information. Indeed, if and to which extent PS implies weak correlations in the


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Figure 3.37. (a) SPP indicator vs. the number n of points falling within V for = 0.01 (triangles, no synchronization), = 0.08 (squares, phase synchronization) and = 0.155 (circles, lag synchronization). (b) SPP vs. n within the lag synchronization regime for = 0.155 (dotted line); = 0.17 (dashed line) and = 0.19 (continuous line). For all cases, before saturation (n 50), the SPP depends on n with a scaling law SPP ∼ nβ with β ∼ 0.85. For n > 50 the three curves saturate to 100 % of synchronization points. Reprinted with permission from Ref. [131]. © 2004 The American Physical Society

chaotic amplitudes is still a matter of debate. The present result shows that PS does imply an increasing percentage of local functional relationship, thus quantifying directly the degree of amplitude synchronization within such a regime. Finally, other novel information can be extracted from the scaling behavior of SPP with n, that is with enlarging the radius δ of the image box in the Y subspace. Figure 3.37(a) shows SPP vs. n for the NS, PS and LS regimes. In all cases, the SPP indicator increases monotonically. For LS (circles) it saturates fast to 1 (the same value as CS). This is reflecting the fact that LS differs from CS only due to the presence of a lag time τ . Enlarging too much the neighborhood size results in V to fully overlap with all images of points in U shifted by a phase factor ωτ , where ω is the mean frequency of the oscillator, thus making indistinguishable LS from CS. More insights on this property can be extracted from Figure 3.37(b), where SPP is reported vs. n within the LS regime for different values of , corresponding

110


to different values of the lag time τ . Here one sees that, before saturation, SPP depends on n with a scaling law SPP ∼ nβ with a unique exponent β ∼ 0.85 for the three values. However, the three curves saturate to 1 at three different values of n, reflecting the behavior of τ within LS, that monotonically decreases as increases. Coming again to Figure 3.37(a), one realizes that for both NS (triangles) and PS (squares), the SPP indicator is always bounded away from 1. This indicates that in these regimes a global predictability of one subsystem’s states from measurement in the other subspace is never possible for any choice of resolution. However, given a resolution δ in the image subspace (a maximum error allowed in the prediction), the proposed indicator quantifies the number of states that can be locally predicted at that resolution, thus revealing that local hidden synchronization features can be extracted for prediction purposes, also in those cases in which global dependencies are not established.

Chapter 4

Effects of Noise

Un día u otro terminará esta illusión, esta vida de la que los dos tratamos de escapar. Y entonces así hará tambien este fragoroso silencio cuyo ruido contundente se extiende hacia esta otra orilla del Mediterraneo, dejando sombras de luz tan dificiles a olvidar.1 In the previous chapters, we have described the emergence of a collective synchronized behavior as a consequence of a coupling between of two (or more) chaotic systems. Another situation that appears in many relevant circumstances is the case in which the systems are uncoupled, or only weakly coupled (that is, the coupling strength is insufficient to setting the emergence of a synchronized motion), but the presence of an additional external noise source may induce or enhance the appearance of synchronization. The presence of noise is, indeed, intrinsic and ubiquitous in nature, where dynamical systems are unavoidably corrupted by external noisy perturbations. The circumstance that different systems are not directly coupled or only weakly coupled but subjected to a common random forcing is of great relevance in biology, in neuroscience and in ecology. For instance, in neural systems, the input signal of a group of different neurons commonly connected to another group of neurons can be conveniently modelled as a noisy source, i.e., as a result of integration of many independent synaptic currents [133]. Another important fact is the experimental observation of the spiking behavior response of some animal neurons to external stimuli. It has indeed been observed that, when the applied stimulus is a constant signal, the neurons generate different sequences of spikes in different repetitions of the experiment. On the contrary, 1 One day or another this illusion will end / this life from which the two of us are trying to escape. / And then also will end this flashy silence / whose forcible noise is getting / into this other side of the Mediterranean sea / leaving shades of light / so hard to forget.

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when a colored Gaussian noise is added to the constant input (which mimics the synaptic input in actual neural systems), the neurons are able to repeat the same spike sequence in each experiments with the same driving signal. Recently, in experiments with lasers, the ability of a nonlinear system to reproduce faithfully (i.e., in successive different trials) a given behavior as the output of a given input signal was called consistency [134]. In ecology, food webs [135,136] and forest ecosystems [137] over a large geographical region are affected by similar environmental fluctuations, and field observations have shown synchronous oscillation of populations [135–137]. Therefore, such environmental fluctuations may have a specific role in setting collective behavior in ecology. At a first glance, one might be tempted to argue that noise has a generic negative effect on synchronization, or that it strongly reduces the stability region for the synchronous motion. And, indeed, if one recalls our discussion on noise induced phase slips on top of a phase locked regime, such a first intuition could be well grounded. But when one deals with nonlinear systems, there are relevant cases in which noise, far from determining negative effects, can give rise to constructive and cooperative effects leading to an enhancement of the synchronization features of the coupled systems, and even it can alone determine synchronization between uncoupled systems. The description of these counterintuitive processes (noise enhanced and noise induced synchronization) is the object of the present chapter, in which we will consider the situation of two chaotic systems that are not coupled directly (or only weakly coupled, i.e., coupled below the synchronization threshold), and therefore whose dynamics would be asynchronous. We will see that a common fluctuating driving signal constituted by a noise source in many contexts is able to enhance (or even induce) synchronization.

4.1. Noise-induced complete synchronization of identical chaotic oscillators Let us start with discussing the simplest case of two uncoupled identical nonlinear chaotic systems subjected to a common drive source. It is obvious (due to the critical dependence of the evolving trajectory on the initial conditions) that, if the systems evolve from different initial conditions, the uncoupled evolution would be strongly asynchronous. Now, let us imagine that the two uncoupled systems are, however, subjected to a common noise source, and let us discuss the issue of how such a common random input could produce synchronization of the systems’ evolution alone (i.e., in the absence of a direct coupling).


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Historically, this problem has sparkled a series of hypotheses and research on the subject. Initially, it was argued that noise could induce synchronization in periodic oscillations [138–140], and later noise-induced order was reported on a chaotic map which was directly connected to the Belousov–Zhabotinshy chemical reaction [141]. In this latter case, a small amount of noise yielded an improvement of the state predictability [142,143], and made that the largest Lyapunov exponent becomes negative. On its turn, it is clear that, if the largest Lyapunov exponent of the noisy driven system is negative, this would imply that an ensemble of such systems with identical laws of motion and common noise, would produce trajectories that shrink in the phase space into a single point [144,145], i.e., it would produce a complete synchronization phenomenon fully induced by noise. In other words, if the system corrupted by noise admits a largest (conditional) Lyapunov exponent which is negative, the system’s evolution would then loose memory of its initial conditions at large times. The effect of a common noise on complete synchronization was studied initially by Ref. [146] in the context of the logistic map xn+1 = 4xn (1 − xn ) + η

(4.1)

being the noise η uniformly distributed in [−W, W ]. η was chosen in a way such that xn+1 ∈ (0, 1). In that study, it was claimed that synchronization can be induced by the common noisy source for W > Wc = 0.5. In the same Ref. [146], furthermore, it was considered also the case of the Lorenz system x˙ = σ (y − x), y˙ = ρx − y − xz, z˙ = −bz + xy,

(4.2)

with noise acting on the y variable which now iterates as √

y(t + t) = y(t) + ρx(t) − y(t) − x(t)z(t) t + W η t. The parameters used in this latter case were σ = 10, ρ = 28 and b = 3/8, and η ∈ (0, 1) was taken as an uniformly distributed biased noise. Also in this case, synchronization was observed for W > Wc 2/3. These preliminary studies, however, soon generated a long-standing discussion. In particular, Ref. [147] pointed out that the largest Lyapunov exponent of the noisy logistic map was positive, indicating that the synchronization effects observed in [146] could be due to some artifact, as the finite precision in numerical simulations [147,148]. The interest was later directed to examine the problem of which specific property should be featured by the noise source to be able to induce synchronization,


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and it was argued that only a noisy source featuring a nonzero mean could eventually lead to synchronization of uncoupled systems, whereas zero-mean noise signals would be unable to induce synchronization [149]. Such a conjecture was further supported by an experimental analysis on synchronization of chaotic Chua’s circuits driven by noise [150], and a general conclusion was drawn that synchronization of uncoupled systems might be induced only by a biased noise, as the result of a noise-induced order generated by the fact that the biased noise drives the system into a noise-smeared periodic orbit. However, the dispute was far from finishing. Indeed, Ref. [151] showed that symmetric white noise can induce synchronization in some chaotic maps, which are chaotic models of coupled neurons [152] featuring significant contraction regions, and synchronization induced by unbiased noise was furthermore reported in other maps with similar exponential tails [151,153]. The controversy was recently clarified in Ref. [154], that carried out a comparative study of noise-induced synchronization in the Rössler and the Lorenz systems. The individual noisy Rössler system reads x˙ = −ωy − z, y˙ = ωx + 0.15y + Dξ(t), z˙ = 0.4 + z(x − 8.5)

(4.3)

with ω = 0.97. As for the Lorenz system with noise, it is described by x˙ = σ (y − x), y˙ = ρx − y − xz + Dξ(t), z˙ = −bz + xy.

(4.4)

The parameters are σ = 10, ρ = 28 and b = 8/3. For both systems, ξ(t) is a delta-correlated Gaussian noise (ξ(t)ξ(t − τ ) = δ(τ )). The largest Lyapunov exponent λ1 of the noisy system can be calculated as a function of the noise intensity. Furthermore, a quantitative measure of complete synchronization is the synchronization error |x1 − x2 | between the two identical systems subjected to a common noise, where here the average is intended both over time and over different noise realizations. The results of λ1 and |x1 − x2 | for the Rössler system and the Lorenz system are shown in Figure 4.1. In the Rössler case, λ1 remains positive until the systems become unstable for D > 4, thus implying that complete synchronization can never be observed. On the contrary, when the Lorenz system is considered, λ1 becomes negative for D > 33.3, and therefore this indicates that two identical Lorenz systems with common noise can achieve complete synchronization, as is further illustrated by a vanishing synchronization error after the transition point.


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Figure 4.1. From Ref. [15]. The largest Lyapunov exponent λ1 of the Rössler system (a) and the Lorenz system (b); and the average synchronization error |x1 − x2 | of the Rössler system (c) and the Lorenz system (d). Notice that, in the Rössler case, λ1 remains positive until the systems become unstable for D > 4, thus implying that complete synchronization can never be observed. At variance, for the Lorenz system, λ1 becomes negative for D > 33.3, and this indicates that two identical Lorenz systems with common noise can achieve complete synchronization.

An important point to remark is the observation that, in the Lorenz case, the typical topological structure of the attractor (the so-called “butterfly” structure) is not changed even for relatively large noise intensities, as it can be appreciated looking at Figure 4.2. At this point, the natural question is why there is such a difference between the Lorenz and the Rössler case, and which are the fundamental reasons for which synchronization by a common noise is achieved in one case, while it is prevented in the other case.

Figure 4.2. From Ref. [15]. The trajectories in the phase space of the Lorenz system at different noise intensity. The dotted background shows the contraction region in the plane y = 0. It has to be remarked that the presence of noise induces an increase of the mean permanence time of the system onto such a contraction region, thus changing the balance in the average between positive and negative contributions to the Lyapunov exponents.


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To answer this question, it is convenient to refer to a generic framework of two identical systems described by x˙ = f (x) + Dξ and y˙ = f (y) + Dξ with x, y ∈ R M . A small difference δx = y − x in the initial conditions will evolve according to the linear dynamics δ x˙ = Df (x)δx where Df (x) stands for the Jacobian matrix. Notice that such a linear equation is the same as for the noise-free case (D = 0), where the system is chaotic, i.e., the maximal Lyapunov exponent 1 λ1 = lim T →∞ T

T ln 0

|δx| |δx0 |

is positive. The only difference between the noise-free case and the noisy case is that the Jacobian matrix Df (x) has to be calculated now onto the noisy trajectories, are different from those emerging in the noise-free case, as they may explore some regions in the phase space which were unreachable by the original chaotic systems. Now, suppose that a given system is characterized by the presence of a large contraction region in the phase space. This is tantamount to say that there exists a very extended region of phase space where all the M eigenvalues of the Jacobian matrix Df (x) have a negative real part, i.e., Re(Λi ) < 0 (i = 1, . . . , M) and nearby trajectories converge to each other. One can imagine that the noise-free systems would explore such a contraction region only marginally, while it would stay much longer in the expansion region of the phase space, so as the corresponding Lyapunov exponent, that is an average over long time trajectories, would result positive. Since now the presence of noise could induce an increase of the mean permanence time of the system onto such a contraction region, the exploration of such a region for larger and larger time intervals would change the balance in the average between positive and negative contributions to the Lyapunov exponents, leading eventually (i.e., for large enough noise intensities) to a change in sign of the maximum Lyapunov exponent. The above reasoning is fully confirmed in the case of the Lorenz system. Here, indeed, the origin (0, 0, 0) is a saddle point embedded in the chaotic attractor, and

4.2. Noise induced phase synchronization of nonidentical chaotic systems

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the chaotic trajectories leaving the neighborhood of this saddle point will come back to its neighborhood. Due to this homoclinic return of chaotic orbits, there exists a large contraction region in the z direction, close to the stable manifold of the saddle point (see Figure 4.2). The presence of noise makes that the system trajectories cannot come back arbitrarily close to the saddle point, and explore deeper and deeper into the contraction region. This obviously changes the competition between contraction and expansion, so as, when the contraction dominates over the expansion, the largest Lyapunov exponent becomes negative and complete synchronization occurs. At variance, in the Rössler system trajectories spiral outwards following the guidance of the two-dimensional unstable manifold of the focus and are fold back by the nonlinearity. A contraction region with all three Re(Λi ) < 0 does exist, but the contraction is very weak because the largest Re(Λi ) is close to zero. In addition, in the presence of noise the system still spend only a small portion of time in the contraction region, and therefore the balance between contraction and expansion is here only marginally affected by noise. In the view of the results of Ref. [154] it is evident that the crucial point is not the biased or unbiased nature of the source. Rather, the ability of noise in inducing synchronization in a system depends on the characteristics of the system itself: what matters is the presence in phase space of the uncoupled systems of a significant contraction region. This means that noise induced synchronization is only possible for a particular class of systems, those ones featuring large contraction regions in phase space, and so allowing for the additional noise to change the balance between contraction and expansion.

4.2. Noise induced phase synchronization of nonidentical chaotic systems The next step is to consider the effects of weaker levels of noise in uncoupled systems that are moreover nonidentical, and show how the addition of such weak levels of noise can induce phase synchronization in a statistical sense. Let us recall that the regime of phase synchronization is associated with the transition of a zero Lyapunov exponent to negative values. We now consider a system subjected to a sufficiently weak noise, and described by x˙ = f (x) + ξ. If the external noise is very small, it is reasonable to assume that, along the dynamical evolution of the system most of the time one has that f (x) |ξ |.


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Therefore, one can approximately speak of a motion along the trajectory and connect the original zero Lyapunov exponent to it. Some recent numerical results, published in Ref. [154], have shown that λ2 can in fact become negative for strong enough noise, and that, in parallel, the phases of two slightly nonidentical oscillatory systems driven by such common noise source become statistically correlated. In particular, Ref. [154] considered two systems with a small parameter mismatch. Particularly, ω1 = 0.97 and ω2 = 0.99 in the Rössler system (4.3) and σ1 = 10, ρ1 = 28 and σ2 = 10.2, ρ2 = 28.5 in the Lorenz system (4.2). In the presence of noise, it is not reasonable to assume phase coherent properties in the dynamics, and therefore the definition of instantaneous phase can be complicated. However, Ref. [154] demonstrated that a properly defined phase variable is in both cases φ = arctan(v/u), where u = x − x0 ,

v = y − y0

for the Rössler system, and u = x 2 + y 2 − x02 + y02 ,

v = z − z0

in the Lorenz system, where (x0 , y0 , z0 ) are the coordinates of the unstable focus point. The second point to be discussed is how to properly monitor the setting of a phase synchronized state. As the evolution of the systems is stochastic, a perfect synchronization of the phases φ1 and φ2 (that would correspond to the condition |φ1 − φ2 | < const) cannot be observed in the absence of a coupling. It is therefore necessary to introduce a statistical approach to conveniently visualize the effects of the noise source. This can be done by computing the distribution of the cyclic phase difference (the modulus 2π of |φ1 − φ2 |) on the interval [−π, π] [155,156]. The appearance of a distribution peaked around a given value would manifest that a preferred phase difference between the systems is statistically selected during the evolution [157]. The situation for the Rössler system is well described in Figure 4.3. Without noise, the two nonidentical Rössler systems are not phase synchronized and the phase decreases almost monotonously (Figure 4.3(a), D = 0). Hence, the distribution of the cyclic phase differences on [−π, π] is very close to a uniform one (Figure 4.3(b)). On the contrary, for strong enough noise we have λ2 < 0, and one observes many plateaus in the phase difference, i.e., many phase-locking epochs (Figure 4.3(a), D = 3.0). In correspondence, one has a pronounced peak around φ = 0 in the distribution of cyclic phase difference (Figure 4.3(c)).

4.2. Noise induced phase synchronization of nonidentical chaotic systems

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Figure 4.3. From Ref. [15]. (a) Time series of phase difference of two nonidentical Rössler systems (ω1 = 0.97 and ω2 = 0.99), at noise intensity D = 0 and D = 3. In this presentation, the phases are lifted (plus 2π for each cycle). The distributions for the cyclic phase differences in the range [−π, π ] are reported for D = 0 (b), and D = 3 (c). Notice that in the former case the distribution is almost homogeneous, while in the latter case a clear peak indicates a statistically preferred value for the phase difference.

Obviously, one has to introduce a proper measure to provide a quantification of the degree of noise-induced phase synchronization. Ref. [154] proposed to calculate the mutual information between the cyclic phase dynamics (on [−π, π]) of the two systems, defined by M1 =

i,j

p(i, j ) ln

p(i, j ) , p1 (i)p2 (j )

(4.5)

where p1 (i) and p2 (j ) are the probabilities when the phases φ1 and φ2 are in the ith and j th bins, respectively, and p(i, j ) is the joint probability that φ1 is in the ith bin and φ2 in the j th bin. For the example of the Rössler system, Ref. [154] took a number N = 100 of bins in the interval [−π, π]. Furthermore, to take into account the change of the individual distributions p1 and p2 due to increasing noise intensities, the mutual information was normalized into [0, 1] as M = 2M1 /(S1 + S2 ), where Sk = − i pk (i) ln pk (i) is the Shannon entropy. The results of mutual information M as a function of noise intensity are shown (along with λ2 ) in Figure 4.4 both for the Rössler and the Lorenz system. One immediately notices that there is not a direct correspondence between the transition point of λ2 (market in the figure by a vertical dashed line) and the point

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Figure 4.4. From Ref. [15]. The second Lyapunov exponent λ2 of the Rössler system (a) and the Lorenz system (b), as functions of the noise intensity D. Panels (c) and (d) report the mutual information M for the Rössler case (c) and the Lorenz case (d). While a direct correspondence between the transition point of λ2 and the point at which phase synchronization is emerging cannot be individuated, one notices that, as λ2 becomes appreciably negative, M increases rapidly, indicating an increasing degree of phase synchronization.

at which phase synchronization is emerging. This is, of course, a direct consequence of the phase incoherence of the stochastic systems. Nevertheless, when λ2 becomes appreciably negative, M increases rapidly, indicating an increasing degree of phase synchronization.

4.3. Noise enhanced phase synchronization in weakly coupled chaotic oscillators The following natural question is what happens in the case of chaotic oscillators that are only weakly coupled (i.e., they are coupled below the threshold for the appearance of phase synchronization). In this section, we will show that the addition of a noisy source can here enhance the primer of phase locking. In order to illustrate the main properties of noise enhanced phase synchronization, Ref. [158] studied two mutually coupled chaotic Rössler oscillators influenced by a common noise: x˙1,2 = −ω1,2 y1,2 − z1,2 + C(x2,1 − x1,2 ),

(4.6)

y˙1,2 = ω1,2 x1,2 + 0.15y1,2 + Dξ(t),

(4.7)

z˙ 1,2 = 0.4 + (x1,2 − 8.5)z1,2 ,

(4.8)

with ω1 = 0.97 and ω2 = 0.99.

4.3. Noise enhanced phase synchronization in weakly coupled chaotic oscillators

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Figure 4.5. From Ref. [15]. Noise enhanced phase synchronization in two weakly coupled Rössler chaotic oscillators (C = 0.0205). (a) Phase difference vs. time for different noise intensities D (indicated in the figure). (b) Average duration of phase synchronization epochs vs. noise intensity D for different coupling strength C. The error bars are only shown for the case C = 0.0205. Note that a log-scale is used in the y-axis in (b).

When the noise is absent, the two oscillators set in a regime of phase synchronization for values of the coupling strength exceeding the synchronization threshold Cps = 0.0208. More precisely, for C < Cps , the phases of the two systems behave in a fully nonsynchronized manner, whereas, as C approaches Cps , the temporal evolution of the phases is characterized by rather long epochs of phase synchronization alternated by phase slips, and accordingly, a pronounced peak in the distribution of the cyclic phase differences emerges. Ref. [158] considered the regime of weak coupling strengths, shortly before the onset of phase synchronization. The corresponding dynamics is illustrated in Figure 4.5(a) for C = 0.0205. In particular, when the common noise is absent (D = 0), the phase difference φ = φ1 − φ2 decreases continuously in time, featuring, however, many epochs of phase synchronization interrupted by phase slips (the typical duration of each phase synchronization epoch is about 300 cycles of oscillations). As soon as one considers the presence of a proper amount of common noise, the duration of the synchronization epochs is strongly enhanced, and the two oscillators are able to maintain phase synchronization for longer and longer periods of time (e.g., for D = 0.1 the mean duration of the synchronization epochs is about 3000 cycles of oscillations, i.e., ten times larger than that in the absence of noise). Such an enhancement is not monotonic with the amount of noise, and, as the common noise becomes strong enough (e.g., D = 0.3), phase slips start again to occur more frequently in the system.

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The most obvious indicator that one can use to characterize quantitatively this phenomenon of noise-enhanced phase synchronization, is therefore the average duration τ of the phase synchronization epochs [158]. As illustrated in Figure 4.5(a), the appearance of very long epochs of phase synchronization is reflected by a distribution of τ that becomes strongly asymmetrical around its mean value τ . As a consequence, one can monitor the manifestation of those very long epochs, by computing the deviation 2 στ2 = τ − τ separately for τ < τ and τ > τ , in order to give a proper measure of the asymmetrical distribution featured at large τ . The average duration itself τ increases with the noise intensity D up to reaching a maximal value, and then decreases for larger D (Figure 4.5(b)). This implies that an optimal amount of noise exists able to enhancing phase synchronization. Such a resonant-like behavior is typical in the interplay between noise and nonlinearity in dynamical systems [159–163] and shares many similarities with the so-called stochastic resonance phenomenon [159], where a bistable system is subjected to an external weak modulation, which is unable by itself to produce jumps from one to the other of the fixed system’s configuration. In stochastic resonance, an additional intermediate noise induces jumps between the two stable configurations at an average frequency equal to that of the weak modulating signal, whereas large noises make that such jumps between the two stable configurations occur erratically. In the case of noise enhanced phase synchronization, the reason for the observed resonant-like behavior is the competition between two underlying effects: from one side the presence of noise induces phase incoherence in the system, from the other side the common noise enhances phase synchronization. In the plot for C = 0.0205 (Figure 4.5, the error bar corresponding to asymmetrical deviation στ shows clearly that common noise induces very long synchronization epochs).

4.4. Constructive noise effects in systems with noncoherent phase dynamics In this final section, we would like to give details of each one of the noise effects discussed in the previous sections, with reference to an experimental system, consisting of a single mode CO2 laser a specific phase noncoherent dynamical regime, called homoclinic chaos [164]. This kind of behavior has the structure of a saddle point S embedded in a chaotic attractor. The trajectories that start from a neighborhood of S leave S along the unstable manifold, and have a fast and close recurrence to S along the

4.4. Constructive noise effects in systems with noncoherent phase dynamics

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stable manifold after a large excursion (spike) in phase space. This implies the existence of a significant contraction region close to the stable manifold. The associated temporal behavior features a sequence of spikes, separated with each other by means of widely fluctuating time intervals T . A relevant property of such a dynamical behavior is that it is highly nonuniform, i.e., the sensitivity to external perturbations is high only in the vicinity of S, along the unstable directions. As a consequence, one may well expect that a weak external noise may influence T significantly. This kind of dynamical system is therefore the ideal candidate for an experimental test and verification of all the noise induced and enhanced phenomena that we have described so far. The experimental demonstration of nontrivial effects of noise was performed in a single mode CO2 laser, both experimentally and numerically, in Ref. [165]. The used experimental setup was a CO2 laser with an intra-cavity loss modulator, driven by a feedback signal proportional to the laser output intensity. By setting suitably the experimental parameters, indeed, the system operates in a regime of homoclinic chaos, where the laser output consists of a chaotic sequence of spikes [166,167]. Specifically, in order to investigate the role of an external noise, a Gaussian noise generator was inserted into the feedback loop, having a high frequency cutoff at 50 kHz. The observed laser dynamics is conveniently represented by the following model [167] x˙1 = k0 x1 (x2 − 1 − k1 sin2 x6 ),

(4.9)

x˙2 = −γ1 x2 − 2k0 x1 x2 + gx3 + x4 + p0 ,

(4.10)

x˙3 = −γ1 x3 + gx2 + x5 + p0 ,

(4.11)

x˙4 = −γ2 x4 + zx2 + gx5 + zp0 ,

(4.12)

x˙5 = −γ2 x5 + zx3 + gx4 + zp0 , (4.13) rx1 x˙6 = −β x6 − b0 + (4.14) + Dξ(t). 1 + αx1 Here, x1 stands for the laser output intensity, x2 for the population inversion between the two resonant levels, and x6 for the feedback voltage signal which controls the cavity losses. The additional variables x3 , x4 and x5 account for molecular exchanges between the two levels resonant with the radiation field and the other rotational levels of the same vibrational band. The parameters of the model are: k0 representing the unperturbed cavity loss parameter; k1 determining the modulation strength; the coupling constant g; the population relaxation rates γ1 , γ2 ; the pump parameter p0 ; the number of rotational levels z; and β, r, α which are, respectively, the bandwidth, the amplification and the saturation factors of the feedback loop.

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The experimentally observed homoclinic chaotic dynamics is fully reproduced by setting the following model parameters: k0 = 28.5714, k1 = 4.5556, γ1 = 10.0643, γ2 = 1.0643, g = 0.05, p0 = 0.016, z = 10, β = 0.4286, α = 32.8767, r = 160, and b0 = 0.1031. On its turn, D is the parameter that controls the noise intensity, and, in the model, D = 0.0005 affects the dynamics of the variable x6 . 4.4.1. Noise-induced changes in time scale and coherence resonance The integration of equations (4.14) shows that, in the absence of external noise, the behavior of the laser output is made of large spikes, followed by a fast damped train of a few oscillations towards S and a successive longer train of growing oscillations spiraling out from S (as it can be seen in Figure 4.6(a)). Figure 4.6(b) shows the trajectory in the space (x1 , x2 , x6 ) manifesting that the damped oscillation features strong contraction along the stable manifold in the phase space, while the growing one corresponds to a weak expansion along the unstable manifold, which can be described approximately as

X(t) ∼ X0 exp λu (t − t0 ) cos ω(t − t0 ), (4.15) where λu ± iω are the eigenvalues of the unstable manifold of S and X0 is the distance from S at any re-injection time t0 . This means that the smaller is X0 , the longer is the time taken to spiral out. Therefore, the model displays a broad range of time scales, and, in analogy with what we already discussed for the unperturbed Lorenz system, it can be classified as a phase noncoherent system. The distribution P (T ) of return times T (shown in Figure 4.7(a)) is characterized by the presence of many peaks. It is reasonable to expect that, as soon as an increasing noise amplitude is acting on the system, the perturbed trajectory cannot come closer to S than the noise level, and therefore it induces a larger X0 , with the consequence that the system spends shorter and shorter times following the guidance of the unstable manifold. Already at small small noise intensities (D = 0.0005), a significant change in the time scales of the model is induced: P (T ) is now characterized by a dominant peak followed by a few exponentially decaying ones (Figure 4.7(b)). The same qualitative distribution of T persists in the whole range D = 0.00005 ÷ 0.002. It should be noticed that the experimental system displays always a small amount of intrinsic noise (equivalent to D = 0.0005 in the model), and therefore it shows a very similar distribution P (T ). As noise becomes larger (i.e., D = 0.01), the fine structure of the peaks is destroyed and the distribution of time scales P (T ) shows now a unimodal peak


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Figure 4.6. Upper panel: Laser output intensity x1 for the noise-free model (4.14). Lower panel: Projection of the corresponding trajectories in the 3D phase space (x1 , x2 , x6 ). In the absence of external noise, the behavior of the laser output is made of large spikes, followed by a fast damped train of a few oscillations towards S and a successive longer train of growing oscillations spiraling out from S. Reprinted with permission from Ref. [165]. © 2003 The American Physical Society

(Figure 4.7(c)). Contemporarily, also the mean value T0 (D) = T t (Figure 4.7, dotted lines) decreases with D. Eventually, when the noise intensity becomes too large, it affects the dynamics not only close to S but also during the spiking, so that the spike sequence becomes fairly noisy. In summary, the most coherent spike sequences is observed at a certain intermediate noise intensity, where the system takes a much shorter time to escape from S after the fast re-injection. In this latter case, the main structure of the spike is preserved, as seen in Figure 4.8(c) and (d), showing the laser output of the model and experimental systems, respectively.


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Figure 4.7. Effects of noise in changing the time scales of the system. (a) D = 0; (b) D = 0.0005; and (c) D = 0.01. The dotted lines shows the mean inter-spike interval T0 (D), which decreases with increasing D. Notice that, already at small noise intensities, a significant change in the time scales of the model is induced, such that P (T ) is characterized by a dominant peak followed by a few exponentially decaying ones. Furthermore, as noise becomes larger, the fine structure of the peaks is destroyed and the distribution of time scales P (T ) shows now a uni-modal peak. Reprinted with permission from Ref. [165]. © 2003 The American Physical Society

In order to quantify the coherence associated to a given spike sequence, one can make use of the parameter R, defined by R = T0 (D)/σT ,

(4.16)

where σT is the standard deviation of P (T ). When D increases, R reaches a maximal value and decreases again (Figure 4.9(a,b)), exhibiting coherence resonance [160], both in the model and experimental systems, which occurs as a consequence of a small noise that changes the time spent in the neighborhood of S along the weak unstable manifold. 4.4.2. Noise-induced complete synchronization Ref. [165] further investigated how the addition of a small noise initially shortens the time spent by the system following the guidance of the unstable manifold, and therefore it reduces the degree of expansion, modifying the competition between contraction and expansion processes, in a way that eventually the former ones become dominant over the latter. In order to quantitatively monitor such an effect, Ref. [165] calculated the largest Lyapunov exponent (LLE) λ1 in the model, as a function of the noise intensity D (Figure 4.10(a), dotted line). One sees, indeed, that λ1 undergoes a transition from a positive to a negative value at Dc ≈ 0.0031. The consequence is that, beyond Dc , two identical laser models x and y with the same noisy driving Dξ(t) would loose the memory of


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Figure 4.8. Time series of output intensities (in arbitrary units) of two lasers’ variables x1 (solid lines) and y1 (dotted lines) in the presence a common noise. Left panels (a,c,e) report the results of the integration of the model equations including independent noise (amplitude D1 = 0.0005 ∼ intrinsic noise level). Right panels (b,d,f) reports the observations of the experimental output signals. In each panel, the value of the common noise intensity is reported. Reprinted with permission from Ref. [165]. © 2003 The American Physical Society

their different initial conditions after a suitable transient time, and would eventually evolve in complete synchronization. This is further confirmed in Figure 4.10(a), solid line, that reports the normalized synchronization error E=

|x1 − y1 | |x1 − x1 |

and shows that E is vanishing beyond Dc .

128


Figure 4.9. The coherence resonance phenomenon in the model (a) and experimental (b) systems. As the common noise intensity D increases, the coherence factor R reaches a maximal value both in the model and in the experiment, as a consequence of the induced changes in the time spent in the neighborhood of S along the weak unstable manifold. Reprinted with permission from Ref. [165]. © 2003 The American Physical Society

Figure 4.10. The noise-induced synchronization (NIS) in the model (a) and experiment (b) systems. In the panel (a), the dotted line reports the largest Lyapunov exponent λ1 , while the solid line reports the normalized synchronization error E between the two fully identical laser models x and y. Finally, the closed circles report the values of E between two identical lasers when a small independent noise (of intensity D1 = 0.0005) is considered, and the open squares refer to the values of E for two slightly nonidentical lasers with b0 = 0.1031 and b0 = 0.1032, respectively. All quantities are plotted as functions of the intensity D of the common driving noise. Reprinted with permission from Ref. [165]. © 2003 The American Physical Society

When, however, the noise becomes too large, also the expansion processes will be significantly affected by it, and will again become important. On its turn, this implies that the largest Lyapunov exponents will start to increase, up to the point


129

at which λ1 crosses again the horizontal axis for D > 0.052, and synchronization will be lost. If one wants to compare the numerical results with the outcome of the experimental laser system, however, there are few complications that must be taken into account. The most important is that, besides the common driving noise, one has necessarily to account for the presence of a second noise source, representing small intrinsic noise fluctuations, that will be different from one to another trial of the experiment. This was modelled in Ref. [165] by explicitly introducing into the equations for the variable x6 a small amount of independent noise (with amplitude D1 = 0.0005) in addition to the common one Dξ(t). The consequence is that the sharp transition to a synchronized regime in fully identical model systems (Figure 4.10(a), solid line) is now smeared out (Figure 4.10(a), closed circles) to a smoother curve. A very similar de-synchronization effect is observed in dependence of a parameter mismatch, as it is shown by the error E calculated for two nonidentical lasers in the homoclinic regime (with b0 = 0.1031 and b0 = 0.1032) subjected to the same random forcing (Figure 4.10(a), open squares). Figure 4.10(b) reports the corresponding experimental curves. Precisely, at each value of the noise intensity D, the experiment was repeated twice with the same external noise. The results are fully consistent with the numerical results in the case of small independent noise: in particular the synchronization error E does not reach zero due to the intrinsic noise, and it increases slightly at large D. One can further gather a visual representation of this process by monitoring (Figure 4.8) the portions of the temporal evolution for the numerical (a,c,e) and experimental (b,d,f) laser intensities at three different amplitudes of the external noise. Precisely, Figure 4.8(a) and (b) correspond to the noise free evolution of the system, Figure 4.8(c) and (d) to the noise value Dmax at which the coherence factor R is maximal, and Figure 4.8(e) and (f) to a noise intensity D ≈ 2Dmax . The scenario depicted in Figure 4.8 gives a direct visual representation of the effects of the common noise on the system: when noise is absent, the two signals are fully unsynchronized; as D ≈ Dmax , both experimental and numerical results demonstrate the existence of almost complete synchronization (the spike sequences show maximal coherence and a smaller average T ); eventually, at a larger noise, synchronization is intermingled with epochs of desynchronization. 4.4.3. Noise-enhanced phase synchronization, deterministic and stochastic resonance Finally, Ref. [165] investigated how the external noise influences the time scales of the dynamics, this way enhancing phase synchronization. This issue, indeed, is

130


of particular relevance in the present case, because the noiseless system displays a phase noncoherent dynamics, and therefore perfect phase synchronization in absence of noise would be hard to be observed. In order to analyze this effect, one can introduce, both in the model and in the experimental system, an external modulation in the pump parameter p0 by a periodic signal with a small amplitude A and a frequency fe . The effective time dependent pump parameter now becomes

p(t) = p0 1 + A sin(2πfe t) .

(4.17)

The noiseless case was reported in Ref. [168], in which it was shown how phase synchronization of the laser output to the modulating signal can be set experimentally. In order to monitor phase synchronization effects, one can compute the phase difference θ (t) = φ(t) − 2πfe t, where the phase φ(t) of the laser spike sequence is defined as t − τk φ(t) = 2π k + (4.18) (τk t < τk+1 ) τk+1 − τk and τk is the spiking time of the kth spike. We have shown previously that the presence of noised induces important changes in the time scales of the dynamics, so as the model responds differently to a weak signal (A = 0.01) with a frequency fe = f0 (D) = 1/T0 (D) (T0 (D) = T t at A = 0), i.e., equal to the mean spiking rate of the unforced model.

Figure 4.11. The response of the laser model to a weak signal (A = 0.01) at various noise amplitudes. The distribution of time scales is now reported for (a) D = 0; (b) D = 0.0005; and (c) D = 0.01. The signal period Te in (a), (b) and (c) corresponds to the mean inter-spike interval T0 (D) of the unforced model (A = 0), respectively. Reprinted with permission from Ref. [165]. © 2003 The American Physical Society


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Figure 4.12. Phase synchronization of laser model to a weak driving signal at various noise intensities. The temporal evolution of the phase difference is reported at various values of D. Reprinted with permission from Ref. [165]. © 2003 The American Physical Society

Figure 4.13. The relative frequency difference Ω (a) and phase diffusion Dθ (b) as functions of the noise amplitude D for various fixed relative initial frequency difference ω. In all cases, A = 0.01. Reprinted with permission from Ref. [165]. © 2003 The American Physical Society

At D = 0, the distribution of time scales P (T ) of the forced model still has many peaks (Figure 4.11(b)), since it corresponds to a phase noncoherent dynamics. As a consequence, phase slips occur frequently and the phase of the laser model is never perfectly locked by the external forcing (Figure 4.12).


132

At D = 0.0005, T is sharply distributed around the signal period Te = T0 (D) (Figure 4.11(b)) and phase slips occur very rarely. At a larger intensity D = 0.01, P (T ) becomes lower and broader again (Figure 4.11(c)) and many random-like phase slips occur. An additional measure of these effects is the dimensionless phase diffusion Dθ as a function of noise amplitude D, namely, Dθ =

2 1 1 d 2 θ (t) − θ (t) , 2πf0 (D) 2 dt

(4.19)

which measures the spreading of an initial distribution of phase difference with the evolution of time. In Figure 4.13, the relative frequency difference Ω and Dθ are reported vs. D for A = 0.01 and for different initial frequency differences ω. One can imme-

Figure 4.14. Stochastic resonance for a fixed driving period. The left panels illustrate the case of the model with A = 0.01, Te = 0.3 ms. The right panels illustrate the experimental measurements, for a forcing amplitude 10 mV (A = 0.01) and period Te = 1.12 ms; here the D is the amplitude of the added external noise. The upper panels show the noise-induced coincidence of the average time scales (dashed line, A = 0) and the synchronization region. The lower panels report the coherence of the laser output. Reprinted with permission from Ref. [165]. © 2003 The American Physical Society


133

diately appreciate the existence of a minimum of Dθ occurring at a certain noise intensity, that is a clear signature of a stochastic resonance phenomenon. The following measure of coherence can be adopted as an indicator of stochastic resonance Te I= σT

(1+α)T e

P (T ) dT ,

(4.20)

(1−α)Te

where and 0 < α < 0.25 is a free parameter. Such an indicator, indeed, accounts for both the fraction of spikes with an interval roughly equal to the forcing period Te and the jitter between spikes. The stochastic resonance of the 1:1 response to the driving signal is evident for both the model and the experiment, in the behaviors of the ratio T t /Te and of I in Figure 4.14. For Te < T0 (0), there exists a synchronization region where T t /Te ≈ 1. The noise amplitude optimizing the coherence I is smaller than that induces coincidence of T0 (D) and Te (dashed lines in Figure 4.14(a,c)). It turns out that maximal I occurs when the dominant peak of P (T ) is located at Te . For Te > T0 (0), noise may induce an n:1 response where the laser produces n spikes per signal period. For example, at Te = 0.6, a 2:1 response can be observed in the laser model which generates 2 spikes with alternate small and large intervals T1 and T2 satisfying T1 + T2 = Te .

Chapter 5

Distributed and Extended Systems

And, for his end, the old man sat. No future to handle, nor past to deny. A present of shadows in the last light of his eyes. No dreams to fulfill, nor fantasies to hide. The smell of the rosemary pervading his mind. No questions to ask, nor answers to find. His thinking abused, his memory dried. That ring twice refused by the only one love of his life. Up to now, we have considered situations in which two (or few) chaotic elements are coupled, in order to describe the conditions for which a collective synchronized motion emerges. In this and the following chapter, we move to discuss how such a collective synchronized motion emerges in large ensembles of coupled dynamical units, or in coupled space-extended fields. As soon as one considers a large ensembles of coupled identical chaotic systems, one has first to distinguish two main types of synchronized motions. The first, that we call global or full synchronization, corresponds to a situation where all the elements of the ensemble display the same behavior in time, independently on the specific initial conditions, and constitutes the natural extension to the case of multiple coupled dynamical units of the synchronization features analyzed so far. The second case, that we call cluster synchronization is, instead, a completely different scenario peculiar of ensembles of coupled elements, and corresponds to the situation wherein the whole ensemble splits into groups of synchronized elements, i.e., groups of elements such that any two elements belonging to the same (to two different) groups are (are not) synchronized. Such a phenomenon was first studied in coupled map lattices [169–178], and in systems of globally coupled maps [170–173,179–181], and then investigated also in locally and globally coupled continuous time chaotic oscillators [182–185]. 135

136

Chapter 5. Distributed and Extended Systems

These two situations will be fully described along this chapter, together with the situation of a coupling between space-extended fields. We furthermore will discuss the specific instabilities that can affect the synchronization manifold in the case of spatially extended systems.

5.1. Synchronization in a chain of coupled circle maps The first report of phase synchronization in ensembles of locally coupled chaotic elements was that concerning the study of chains of diffusively coupled Rössler oscillators [186]. We start here, instead, by describing the main synchronization features emerging in chains of coupled circle maps, since they can be considered as the discrete in time analog of the phase equations describing the behavior of ensembles of coupled continuous in time limit-cycle oscillators. Such systems, indeed, are very similar to phase equations usually obtained for ensembles of coupled limit-cycle oscillators and can be written in the form [6]: φ˙ n = ωn + F (φn ) + d sin(φn+1 − φn ) + sin(φn−1 − φn ) ,

n = 1, . . . , N,

(5.1)

where φn (ωn ) is the phase variable (the natural frequency) of the nth element, N stands for the number of elements, F (φn ) is a nonlinear function responsible for the nonuniformity of the rotations of the nth element. Notice that equations (5.1) do not account for the amplitude nor for other variables that can influence the phase dynamics, and that, for uncoupled elements, the corresponding equations (d = 0 in (5.1)) do not possess a chaotic behavior. Such disadvantages can be partially avoided by considering the following discrete analog of equation (5.1): φnk+1 = ωn + φnk − F φnk k k + d sin φn+1 − φnk + sin φn−1 − φnk , n = 1, . . . , N. (5.2) Here φnk is the phase variable at times k = 1, 2, . . . . The parameter ωn characterizes the partial frequency. The function F (φn ) is taken as having a piece-wise linear form F (φn ) = cφn /π. This function is therefore defined in the interval [−π, π], and c is the control parameter. In the following, we will describe the results of the treatment of the system (5.2) k k. = φN when subjected to free-end boundary conditions, i.e., φ0k = φ1k and φN+1 In this approach, each single element in the chain is represented by a circle map. Since one is interested in the synchronization properties of such chaotic elements, one has to analyze negative values of c. In order to test for m1 : m2 phase synchronization (being m1,2 integer numbers), one can make use of two criteria.


137

The first consists in defining m1 : m2 phase synchronization between two coupled maps as the fulfilment of the phase entrainment or locking condition m1 φ k − m2 φ k < const, (5.3) n n+1 for all k = 1, 2, . . . . A second (weaker) criterion is based on the equivalence of the rotation numbers, i.e., the fulfilment of the condition m1 ρn = m2 ρn+1 .

(5.4)

Here ρn is the rotation number of the nth map, defined as: φ M − φn1 1 lim n , (5.5) 2π M→∞ M where M is the number of iterations. The fulfilment of the conditions (5.3) and (5.4) for all n = 1, . . . , N implies, indeed, the existence of global synchronization. On the contrary, when these conditions are satisfied only for several neighboring elements, a regime of cluster synchronization is formed. To better illustrate the situation, one can make reference to a series of numerical simulations with a chain of 50 elements with linear: ρn =

ωn = ω1 + (n − 1),

n = 1, . . . , N,

(5.6)

and random distribution of the individual frequencies ωn = ω1 + ξn ,

n = 1, . . . , N,

(5.7)

where = const and ξn are uniformly distributed random numbers in the interval [−0.5; 0.5]. One can calculate the rotation number ρn for each CM, and one finds that, in analogy with the self-synchronization in chains of periodic oscillators and chains of chaotic phase coherent Rössler oscillators [186], mutual global synchronization in chains of coupled maps can appear or vanish in two ways: namely by a soft transition or a hard transition. The case of soft transition corresponds to the absence of cluster formation, and is characterized by a smooth and continuous process of locking of the rotation numbers. Such a process can be observed in chains where the frequency mismatch between elements is very small. On the opposite, the hard transition to synchronization corresponds to an appearance (or disappearance) of a global synchronization state accompanied by the existence of cluster synchronization. Such a latter situation generally happens in rather long chains with relatively large frequency mismatch. In this second case, the loss of global synchronization leads to the appearance of two (or many) clusters of elements which rotate at the same rotation number.

138


Figure 5.1. From Ref. [15]. Hard (a) and soft (b) transitions to global chaotic phase synchronization for a chain of N = 50 coupled circle maps (equation (3.38)). The relative rotation numbers ρn /ρ1 at different coupling coefficients d for linear distribution of individual frequencies, b1 = 0.6, frequency mismatch b = 0.002, are reported vs. n. c = −0.002 (a) and c = −0.4 (b).

With a further increase of the frequency mismatch the formation of new clusters is typical. In the soft transition, most elements of the chain (except, perhaps, the edge ones) rotate with different rotation numbers after the loss of global synchronization. When one sets the original systems with a linear distribution of the individual frequencies, the rich spatio-temporal dynamics of the noncluster (smooth distribution of rotation numbers) (Figure 5.1(a)) and cluster synchronization structures (Figure 5.1(b)) is illustrated in Figure 5.2. In all figures, the darker regions correspond to higher values of the variables. The two left panels show the quantity sin(φnk ), so that the white stripes correspond to phases ≈ 3π/2 and the black stripes to phases ≈ π/2. The right panel shows the quantity k − φnk φ sn = sin2 n+1 (5.8) 2


139

Figure 5.2. Reprinted with permission from Ref. [15]. Space (horizontal) – time (vertical) plots of the evolution of sin(xnk ) (a,b) and sn (c,d) during hard (a,c) and soft (b,d) transitions to global chaotic phase synchronization for a linear distribution of individual frequencies. The signals are coded in a grey scale, with minimal values being represented by white and maximal by black. The parameters are N = 50, b1 = 0.6, b = 0.002, d = 0.39. c = −0.002 (a,c); c = −0.4 (b,d).

which characterizes the instantaneous phase difference between neighboring oscillators. One has sn = 0 if the phases are equal and sn = 1 if they differ by π. The spatio-temporal behavior of the boundaries between clusters corresponds to the positions where phase slips or defects occur. These defects are clearly seen as


140

maxima (black regions) of sn , and they can follow regularly in time at certain positions in the chain; this case corresponds to the existence of strong jumps between clusters (Figure 5.2(c)). If the cluster structures do not exist or the borders between them are smooth, then defects appear irregularly in both space and time (Figure 5.2(d)).

5.2. Phase synchronization phenomena in a chain of nonidentical phase coherent oscillators The next step consists in describing the case of a chain of continuous in time oscillators. This situation is conveniently illustrated by presenting the results obtained on a chain of nonidentical Rössler oscillators with a nearest-neighbors diffusive coupling [186]. The model of this chain can be written as a set of ordinary differential equations: x˙n = −ωn yn − zn , y˙n = ωn xn + ayn + ε(yn+1 − 2yn + yn−1 ), z˙ n = 0.4 + (xn − 8.5)zn .

(5.9)

In the above equations, j = n, . . . , N denotes the index of the oscillator in the chain, ωn stands for the natural frequency of the individual oscillator, and ε is the coupling coefficient. In analogy with the case of a chain of coupled maps, one can here introduce the gradient distribution of natural frequencies ωj = ω1 + δ(j − 1), where δ is the frequency mismatch between neighboring systems. A possible variant is a random distribution of natural frequencies in the range [ω1 , ω1 + δ(N − 1)]. The boundary conditions are assumed to be free-end: y0 (t) = y1 (t);

yN+1 (t) = yN (t).

(5.10)

As we discussed in previous chapters, the definition of instantaneous phase for the Rössler system (as long as parameters are selected so as the corresponding attractor is phase coherent) can be taken as φn = arctan(yn /xn ), whereas the corresponding amplitude can be defined as An = xn2 + yn2 .

(5.11)

(5.12)

5.2. Phase synchronization phenomena in a chain of nonidentical phase coherent oscillators 141

As far as we limit ourselves to the case of phase coherent oscillators, the phase of the chaotic system is well defined, and therefore also the phase difference between neighboring oscillators φn − φn+1 can be easily evaluated. In this context, 1:1 phase locking corresponds to a phase difference that does not grow with time, but remains bounded. As a weaker condition of synchronization, one can also consider the equivalence of the averaged partial frequencies defined as: Ωn = = lim

T →∞

φn (T ) − φn (0) . T

(5.13)

The mean frequency of chaotic oscillations Ωj can be also calculated as Ωn = lim 2π T →∞

MTn , T

(5.14)

where MTn is the number of rotations of the phase point around the origin during time T . Such a method can be directly applied to the observed time series, when one, e.g., takes for MTn the number of maxima of xn (t). For the Rössler attractor, the estimates (5.13) and (5.14) practically coincide. Ref. [186] reported the result of numerical simulations performed with chains of 20–50 oscillators, for different values of the parameters δ, ω1 , ε, with a special attention to monitor the observed frequencies Ωn . As one can expect, all the frequencies Ωn become equal with increasing coupling strengths, meaning that a global synchronization state sets in. But, in analogy with the case of coupled periodic oscillators and coupled chaotic circle maps, such a regime of global synchronization in the chain (equation (5.9)) can emerge in two distinct ways, depending on the relatively frequency mismatch δ/ω1 . Two scenarios, referred to as the soft and the hard transitions, are described in Ref. [186]. In particular, for relatively small frequency mismatches δ/ω1 1, a soft transition to global synchronization is observed (Figure 5.3), where the amplitudes of the oscillators remain chaotic, which is clearly marked by the spectrum of Lyapunov exponents (Figure 5.4). Instead, as relatively large frequency mismatch δ/ω1 are taken into account, the transition to global synchronization occurs via progressive clustered states, i.e., a hard transition takes place (Figure 5.5). In this latter situation, as ε increases, the clusters of mutually synchronized oscillators appear rather abruptly, whereas a further increase of coupling induces the width of the clusters to grow in parallel, and the number of clusters to decrease, up to a point at which eventually only one cluster remains, indicating the setting of a global synchronization state.

142


Figure 5.3. Reprinted with permission from Ref. [15]. The soft transition to global synchronization in a chain of Rössler oscillators. Mean frequencies Ωn vs. n, for different values of coupling ε. The parameters are: N = 20, δ = 2 × 10−4 and ω1 = 1.

5.3. Collective phase locked states in chains of phase coherent chaotic oscillators Another interesting case was reported in Ref. [188], and consists in the emergence of phase locked collective states in an open chain of unidirectional coupled chaotic oscillators. To exemplify the situation, Ref. [188] referred to a system composed by N = 11 coupled nonidentical three dimensional Rössler oscillators, described by x˙1 = −ω1 y1 − z1 , y˙1 = ω1 x1 + ay1 , z˙ 1 = f + z1 (x1 − c), x˙i = −ωi yi − zi + ε(xi−1 − xi ), y˙i = ωi xi + ayi , z˙ i = f + zi (xi − c), where i = 2, . . . , N represents the index of the oscillator.

(5.15)


Figure 5.4.

143

From Ref. [15]. The forty largest Lyapunov exponents λn vs. n for three of the coupling regimes reported in Figure 5.3.

Furthermore, in equations (5.15), a = 0.15, f = 0.4 and c = 8.5 are fixed parameters, while the frequencies ωi of the N oscillators increase linearly by the rule ωi = ω1 + (i − 1)

ωN − ω 1 , N −1

(5.16)

where ω1 = 0.985 (ωN = 1.0165) is the frequency of the first (the last) oscillator in the chain. Due to the unidirectional nature of the coupling, ω1 will be the driving frequency of the chain. Ref. [188] monitored the temporal evolution of phase differences among different oscillators ϕij (t) ≡ ϕi (t) − ϕj (t) and verified that ϕij (t) fulfills the locking condition |ϕij | < const. The election of parameters of system (5.15) determines that the attractor, where the trajectory evolves, is phase coherent, and then it is possible to define the phase

144


Figure 5.5. Reprinted with permission from Ref. [15]. The hard transition to global synchronization in a chain of Rössler oscillators. Mean frequencies Ωn vs. n for different values of coupling ε (reported in the figure). The parameters are: N = 50, δ = 9 × 10−3 and ω1 = 1.

of each chaotic oscillator as yi (t) ϕi (t) = arctan xi (t) and the corresponding amplitude as Ai (t) ≡ xi2 (t) + yi2 (t). In particular, one can monitor the phase difference between each coupled oscillator and the first one ϕi1 (t) at different coupling values ε. Ref. [188] reported that, at very small coupling strengths, all oscillators evolve in a unsynchronized manner. The observed phase difference evolutions align quasi perfectly on straight lines whose slopes correspond to the mean frequency differences ωi1 ≡ |−|. As a consequence, each chaotic oscillator evolves with a different rhythm, and no phase locking is produced.


145

As the coupling increases, system (5.15) Ref. [188] gave evidence of a transition toward a collective state, wherein some oscillators (i = 2, 3, 4, 5) display phase locking with the drive frequency ω1 , whereas all the other oscillators evolve in a phase unsynchronized regime. In order to give evidence of phase locking, long simulation trials were performed in Ref. [188], in which the system was prepared in the unsynchronized regime (ε = 0.0015). Initially, it was observed that all oscillators evolve in a phase unsynchronized manner. At t = 5000 time units, a sudden change in the coupling value was realized, and the coupling strength was set at ε = 0.06 in system (5.15). The observed effect is that, after a very short transient time, all oscillators begin evolving in a phase locked regime, and consequently all phase differences converge to constant values. To characterize more quantitatively the observed synchronization scenario, one can study the changes of Lyapunov spectrum as function of the coupling. In the present case, the Lyapunov spectrum at ε = 0 is constituted by N positive, N zero and N negative exponents, where the N zero exponents are associated to the phases of the chaotic oscillators. The results shown in Ref. [188] have shown that, in the uncoupled case (ε = 0), the Lyapunov spectrum is constituted by N = 11 positive exponents (λ+ ), N zero exponents (λ0 ) and N negative exponents (λ− ). As the coupling strength increases, the values of the largest N exponents in the spectrum are modified. Precisely, all N exponents remains positive in the range 0 < ε 0.027. At larger couplings, one exponent passes to a negative value. This implies that no amplitude correlations are built within the system for ε 0.027. As for the values of λ0 (exponents that were 0 in the uncoupled case), it happens that successive exponents passes from zero to a negative value, as ε increases, until ε 0.027. This is an important fact, that indicates the presence in the system of a regime (0 < ε < 0.027) where successive oscillators lock their phases with the drive phase without building appreciable correlations in the corresponding chaotic amplitudes. Finally, Ref. [188] reported that the values of the three largest negative exponents in the Lyapunov spectrum are not changed in sign, as the coupling increases. The behavior of the Lyapunov spectrum helps one to understand quantitatively the scenario of synchronization emerging in the system. For sufficiently small coupling strength, the oscillators evolve in a unsynchronized way. At intermediate coupling strengths, the system displays again N positive Lyapunov exponents in the spectrum, whereas some of the exponents that were originally zero have passed to negative values. As a results, the collective dynamics of system (5.15) displays no correlation in the chaotic amplitudes of the oscillators, but partial locking in the phases. Finally at large coupling strength, the system evolves in a completely phase locked regime.

146


However, this completely phase locked regime cannot be realized without having at least partial correlation in the amplitudes. This is due to the fact that the ε value at which the last oscillator locks its phase with the drive (ε 0.052) corresponds to a situation where at least one originally positive Lyapunov exponent has passed to a negative value, thus indicating the settings of amplitude correlation in at least a pair of chaotic oscillators in the chain. This fact differentiates the present case from the scenario of synchronization phenomena occurring in a pair of chaotic oscillators. Here, indeed, a strict phase locked regime cannot be realized over the whole chain, without implying at least partial correlation in the chaotic amplitudes.

5.4. In phase and anti-phase synchronization in chains of homoclinic oscillators A further step in the understanding of synchronization phenomena in chains of coupled oscillators was made by Ref. [189], that studied the emergence of synchronized motion in chains of coupled homoclinic sites. Along the present and the following sections, we will summarize the main results concerning the emergence and competition of synchronization domains for a specific model system, that corresponds to the experimental behavior of the CO2 laser already presented in Chapter 4. More precisely, this section will focus on the case of a chain of unidirectionally coupled homoclinic oscillators, whereas the next section will discuss the case of a bidirectional coupling involving each element in the chain. We recall what we already discussed in Chapter 4, i.e., that homoclinic chaos is a peculiar dynamical behavior, whose signal (Figure 5.6(a)) is made of a train of spikes. The corresponding 2D phase space projection (Figure 5.6(b)) shows that, in each cycle, the intensity returns to its zero value baseline (point O in Figure 5.6(b)) which represents a saddle node (SN) fixed point, and then it leaves it by means of a large spike followed by a decaying spiral toward a saddle focus (SF). The escape from SF is represented by a growing oscillation, which appears as a chaotic tangle in Figure 5.6(b). Notice that the chaotic region around SF is very contracted in phase space but stretched time-wise, vice versa the spike occurring when leaving SN takes a short time but is spread over a wide space region. The chaotic characteristic of the inter-spike interval (ISI) is due to the chaotic permanence time ts around the saddle focus SF, whereas the permanence time to on the baseline is a fixed refractory time corresponding to the heteroclinic connection to SN; it stabilizes the orbit away from the saddle focus. In this representation, the useful signal is S(t) = S0 i δ(t − τi ), where τi is the time of occurrence of the ith spike, and (ISI)i = τi − τi−1 . In these conditions, a delayed feedback can stabilize complex periodic orbits of period T . These orbits consist of a pseudo-chaotic train of pulses, that is, of a


147

Figure 5.6. (a) Time intensity in the homoclinic regime, taken from experiments with a CO2 laser. The dashed line corresponds to the baseline. The chaotic characteristic of the signal is due to the chaotic permanence time ts around the saddle focus, whereas the permanence time to on the baseline is a fixed refractory time. (b) The corresponding phase space projection of the experimental time series. Reprinted with permission from Ref. [189]. © 2003 The American Institute of Physics

limited sequence with chaotic ISIs, that repeats after a time T again for ever. Of course T is chosen much larger than where is the ISI averaged over a long sequence. It is important to establish under what conditions a chain of such coupled oscillators can give rise to synchronous motion. Since Ref. [189] was interested to establish a comparison with the case of a delayed feedback, and since the delay implies that the information propagates in one direction, just unidirectional coupling have been considered in the oscillators chain, so that the oscillator at the site i is driven by the previous one at the site i − 1. In addition, the delayed reentry of the signal in the system means that the system is exposed to the total information generated over a previous time stretch of size Td . Meeting this condition imposes a closed boundary with the last oscillator coupled to the first one. With these boundary and coupling constraints, Ref. [189] built the array by using the scaled equations x˙1i = k0 x1i x2i − 1 − k1 sin2 x6i , x˙2i = −γ1 x2i − 2k0 x1i x2i + gx3i + x4i + p,


148

x˙3i = −γ1 x3i + gx2i + x5i + p, x˙4i = −γ2 x4i + zx2i + gx5i + zp, =

−γ2 x5i

+ zx3i

(5.17)

+ zp, i r(x1 − x1i−1 ) i i x˙6 = −β x6 − b0 + , 1 + αx1i x˙5i

+ gx4i

where the index i denotes the ith site position, and for each oscillator x1 represents the laser intensity, x2 the population inversion between the two resonant levels, x6 the feedback voltage which controls the cavity losses, while x3 , x4 and x5 account for molecular exchanges between the two levels resonant with the radiation field and the other rotational levels of the same vibrational band. The coupling on each oscillator was realized by adding a function of the intensity (x1 ) of the previous oscillator to the equation of its feedback signal x6 . The parameters are the same for all elements of the chain. Here, k0 is the unperturbed cavity loss parameter, k1 determines the modulation strength, γ1 , γ2 , g are relaxation rates, p0 is the pump parameter, z accounts for the number of rotational levels, β, b0 , r, α are respectively the bandwidth, the bias voltage, the amplification and the saturation factors of the feedback loop, and is the coupling strength. The values used in Ref. [189] for the numerical simulations were: k0 = 28.5714, k1 = 4.5556, γ1 = 10.0643, γ2 = 1.0643, g = 0.05, p0 = 0.016, z = 10, β = 0.4286, α = 32.8767, r = 160, b0 = 0.1032. In these conditions, the coupling strength is the control variable and it can assume both negative and positive values, as in the experiments. If positive, the signal tends to reach in-phase synchronization with the delayed modulation, while if it is negative, the coupling comes out to be phase-repulsive and the signal is in anti-phase with the feedback perturbation. The experimental data [190] reveal a small time offset between the modulation and the signal, which is independent of the long delay Td and of the coupling strength. This offset depends on the sign of the modulation feedback, and it has been measured to be τn = 140 µs for negative coupling and τp = 20 µs for a positive, against an average ISI of 500 µs. The numerical results of Ref. [189] are shown in Figure 5.7, and explain how these asymmetries depend on the coupling sign. In both cases (positive (a) and negative (b) coupling), the slave intensity lags with respect to the driving one, but by fixed amounts, which however differ for different coupling signs. Precisely, the driver’s negative slope forces the slave to escape away from the saddle region toward the zero baseline. In case (a), the driver’s and slave’s collapses onto the zero baseline occur just one after the other and the negative oscillations around the spike do not induce a transition insofar as they occur while the slave is already on the baseline. The offset τp depends on the coupling strength, that is, on how effective is the forcing to let the slave escape from SN.


149

Figure 5.7. Detail of the time profile of the ith (solid line) and (i − 1)th elements of the chain, for (a) positive coupling and (b) negative coupling. The experimental data reveal a small time offset between the modulation and the signal, which is independent of the long delay Td and of the coupling strength. This offset depends on the sign of the modulation feedback, and it has been measured to be τn = 140 µs for negative coupling and τp = 20 µs for a positive coupling. Therefore, in both cases, the slave intensity lags with respect to the driving one, but by fixed amounts, which however differ for different coupling signs. The times τp and τn correspond to different information propagation velocities along the array, with a lower velocity (longer offset) for the negative coupling. Reprinted with permission from Ref. [189]. © 2003 The American Institute of Physics

On the contrary, in case (b) the first negative driver’s slope which can force the slave to fall away from the saddle coincides with the first negative spike oscillation; the escape of the slave from SF toward SN takes a lag time depending on the coupling strength; adding this lag to t0 makes a total offset τn > τp . Therefore we can say that the times τp and τn correspond to different information propagation velocities along the array, with a lower velocity (longer offset) for the negative coupling. In order to model the delayed system with an array of N coupled systems, Ref. [189] tried to match the overall delay along the closed chain N τj (where j

150


Figure 5.8. Different responses of the chain for negative (Nn) and positive (Np) coupling. The overall propagation velocities Nj /T (j = n, p) are in the ratio 1/7 as the two offset times in Figure 5.7. Reprinted with permission from Ref. [189]. © 2003 The American Institute of Physics

stays for p or n depending on the coupling) with a delay time Td so that each system is exposed to a delayed version of its own signal x1 (t − Td ) = x1 (t − N τj ). As τp < τn , a longer chain will be needed in the positive case to obtain the same periodicity. The scale of the different propagation velocities can be seen in Figure 5.8 where the relation between the repetition time T and the number N of oscillators in the chain are reported for the two conditions. In order to adjust to the experiment, Ref. [189] had to choose different coupling strengths for the positive and the negative case, respectively = 0.15 and = −0.042. An example of the dynamical characteristics can be appreciated in Figure 5.9. Here, the time intensity profile of a single site of the chain is compared with its driver neighbor in the pseudo-chaotic regime, for both positive (Figure 5.9(a)) and negative (Figure 5.9(b)) coupling. The chains have been chosen to have approximately the average period = 0.95 ms, and = 1.25 ms respectively, with N = 7 in the negative case and N = 70 in the positive case. One can therefore establish a full equivalence between the experimental laser system of Ref. [190] and the closed chain reported in Ref. [189].


151

Figure 5.9. Temporal evolution of the intensity profiles for two neighbor oscillators. In panel (a) N = 70, = 0.15; in panel (b) N = 7, = −0.042. The data have been furthermore vertically shifted for a better visualization. Reprinted with permission from Ref. [189]. © 2003 The American Institute of Physics

5.5. Synchronization domains and their competition An even richer scenario is observed for a bidirectional coupling in the same chain of HC units. In particular, in this latter case, the issue is to investigate how different synchronization domains can emerge, coexist and/or compete among them. Ref. [191] approached this latter scenario, and considered a one dimensional chain of sites, each one undergoing a local homoclinic chaotic dynamics, interacting via a bidirectional nearest neighbor coupling. Now, given the bidirectional nature of the coupling, a relevant problem emerges related to the ability of the array to respond to external periodic perturbations localized at one end site, yielding synchronized patterns.


152

The issue of the emergence and competition of synchronization domains was addressed in Ref. [191], that referred to a chain of dynamical units, each one modelling homoclinic chaos. The extension of the model already used in the previous section to the case of bidirectionally coupled elements in the chain is: x˙1i = k0 x1i x2i − 1 − k1 sin2 x6i , x˙2i = −γ1 x2i − 2k0 x1i x2i + gx3i + x4i + p, x˙3i = −γ1 x3i + gx2i + x5i + p, x˙4i = −γ2 x4i + zx2i + gx5i + zp, x˙5i = −γ2 x5i

+ zx3i

+ gx4i

i i x˙6 = −β x6 − b0 + r

(5.18)

+ zp, x1i

1 + αx1i

+

x1i−1

+ x1i+1

.

− 2<x1i >

Here, again, the index i denotes the ith site position (i = 1, . . . , N ), and dots denote temporal derivatives. In Ref. [191], the considered units were identical, with parameters equal to: k0 = 28.5714, k1 = 4.5556, γ1 = 10.0643, γ2 = 1.0643, g = 0.05, p0 = 0.016, z = 10, β = 0.4286, α = 32.8767, r = 160, b0 = 0.1032. The coupling on each site is realized by adding to the x6 equation a function of the intensity (x1 ) of the neighboring oscillators. The term <x1i > represents the average value of the x1i variable, calculated as a moving average over the whole evolution time. The authors of Ref. [191] first studied the emergence of synchronization in the absence of external stimuli, as the coupling strength increases. Due to the coupling, a spike on one site induces the onset of a spike in the neighbor sites. The transition to phase synchronization is anticipated by regimes where clusters of oscillators spike quasi-simultaneously [192]. Clusters are delimited by “phase slips” or defects, easily seen as holes in the space time fabric. More precisely, one can introduce a phase measure φ i (t) for a time interval t i , τ i , by linear between two successive spikes of the same site, occurring at τk−1 k interpolation: φ i (t) = 2π

i t − τk−1 i τki − τk−1

.

(5.19)

A defect appears as a 2π “phase slip” in the difference between the phases of two adjacent sites. The size of the synchronization cluster increases with , extending eventually to the whole system.


153

The route to phase synchronization can be characterized by the defect density, that is, the number of defects per site, as a full phase synchronization state would correspond to have a defect density below one defect per site. The defect statistics has been studied for several chain lengths by Ref. [191], that found that above N = 30 there are no appreciable size-dependent effects. Once phase synchronization is established, a further increasing of reduces the natural frequency of the system 2π . () It is in these conditions that Ref. [191] further studied how such a slowing down affects the capability of the array to synchronize to an external signal. For this purpose, one can explore the response of the system to an external periodic stimulus applied to the first site of the chain. Precisely, one periodically modulates the parameter bo at the site i = 1 as bo1 = bo (t) = bo 1 + A sin(ωt) . ωo () =

As it is known that this kind of driving can induce a phase synchronization on a single oscillator, it is therefore relevant to understand how the synchronization state can propagate all throughout the chain, determining the ability of the system to propagate the periodic signal up to the other end of the chain. The first observation of Ref. [191] was that the modulation amplitude A does not affect significantly the results, provided that it is sufficient to synchronize the i = 1 site, allowing one to fix it once forever at a constant value A = 0.3. As a measure of the ability of the system to propagate the synchronous state, one can use the following criterion: the signal has been successfully transmitted through the system when, after a finite time, the last element of the chain spikes with the same period of the external forcing, without defects. In Figure 5.10(a)–(b) examples are given of partial signal transmission. If ω is too small (Figure 5.10(a)) or too large (Figure 5.10(b)) with respect to the natural frequency of the system (ωo () = 0.02 in the figure), then only partial transmission is achieved. One can then explore the (, ω) range over which transmission propagates over the whole chain. In the low frequency limit, one find that ω < ωo () is not able to globally synchronize the chain. Independently of N, as ωo is larger than ω, the last sites tend to spike spontaneously between two consecutive periods of the external driver before the synchronization propagates to them, and therefore synchronization is lost (Figure 5.10(a)). When ω > 2ωo (), the first Np sites synchronize with the driving frequency, but beyond Np a line of defects restores the natural oscillation regime (Figure 5.10(b)). In Figure 5.10(c) the penetration depth Np is reported vs. the forcing frequency for different values of .

154


Figure 5.10. Time response of the N = 40 chain with = 0.13 and ωo () = 0.02 to an external periodic forcing: (a) ω = 0.015, (b) ω = 0.042. (c) Penetration depth vs. ω for different coupling strengths =: 0.12 (∗), 0.15 (e), 0.2 (1), 0.25 (!). Reprinted with permission from Ref. [191]. © 2003 The American Physical Society

If for a point (, ω) the penetration depth is Np , then one would observe complete synchronization only for sizes N < Np , while incomplete synchronization would unavoidably take place for N > Np . As a result, for a given N , only a limited range of external frequencies can be transmitted over the whole chain. In Figure 5.11 the boundaries of the transmission band are reported as a function of and ω, for several chain lengths. The region inside the curves contains all the (, ω) points for which global transmission is allowed. It can be seen that for each , the transmission band extends from ωo (black stars) to approximately 2ωo . Notice that the system starts to transmit for coupling strengths above the ones leading to intrinsic synchronization (approximately > 0.11). The left boundary of the transmission range refers to perfect transmission of the ω period up to the end of the chain. If one is only interested in the transmission of the average frequency, this boundary is slightly smeared out. With the help of the above information, one has all the necessary tools to address the main question of how two different frequencies (applied at the two ends of the chain) compete in generating separate spatial patterns of synchronization. This problem (the temporal competition between different synchronization states) had been investigated both theoretically [193] and experimentally [194] in the context of a single chaotic system forced by two external frequencies, finding


155

Figure 5.11. The curves delimiting the (, ω) range for which synchronized transmission occurs in the arrays. The different curves refer to different array’s sizes: N = 10 (dotted lines), N = 20 (dotted-dashed lines), N = 40 (dashed lines) and N = 80 (solid lines). The black stars indicate the average spiking frequency ωo () of a site in the array, when the external perturbation is absent. The inset is a zoom of the area where the two external frequencies are selected for the analysis of the spatial competition between synchronization domains. Reprinted with permission from Ref. [191]. © 2003 The American Physical Society

a series of interesting competitive behaviors, such as alternations of synchronism to several frequencies (ω1 , ω2 or a combination of the two). In order to investigate the very same question but in the case of a spatial chain of coupled oscillators, Ref. [191] applied to the first (i = 1) and last (i = N ) site two periodic perturbations with frequencies ω1 and ω2 , respectively, selected such that ωo < ω1 < ω2 . The consequence is Np (ω1 ) > Np (ω2 ), that is the penetration depth of the first perturbation is strictly larger than the penetration depth of the second perturbation (see Figure 5.10(c)). The emerging competition scenario can be described with reference to Figure 5.12. For Np (ω1 ), Np (ω2 ) > N, both frequencies synchronize over the whole chain. However, after a suitable transient time, the whole system synchronizes to the larger frequency ω2 , with the only exception of the site i = 1 (Figure 5.12(a)).

156


Figure 5.12. Competition between spatial synchronization regimes induced by external forcing: (a) ω1 = 0.02, ω2 = 0.021, = 0.13; (b) ω1 = 0.038, ω2 = 0.042, = 0.12; (c) ω1 = 0.04, ω2 = 0.0405, = 0.11. Notice that, for Np (ω1 ), Np (ω2 ) > N , both frequencies synchronize over the whole chain, with the larger frequency ω2 dominating in the asymptotic state. For Np (ω1 ) > N , Np (ω2 ) < N , only the smaller frequency synchronizes over the whole chain, while the larger frequency is limited to the Np (ω2 ) sites closest to i = N . In this situation, one finds that permanent synchronization domains for ω1 and ω2 are established, with a domain wall, that evolves irregularly in space and time (b). Finally, for Np (ω1 ), Np (ω2 ) < N , neither of the two frequencies stabilizes a synchronized pattern over the whole chain, and one observes alternation between synchronization patterns with frequencies ω1 and ω2 , with intervals of asynchrony filled with defects. Reprinted with permission from Ref. [191]. © 2003 The American Physical Society

This is due to the fact that a wave with a higher frequency diffuses along the space faster than a low frequency one. As a result, the wave coming from the ω2 source arrives first to induce a spike in the ith site, while the ω1 wave arrives to the same point when the oscillator is not receptive to the signal. Therefore, the ith site will oscillate at the ω2 frequency and will induce the next site to spike at the same frequency. For Np (ω1 ) > N , Np (ω2 ) < N , only the smaller frequency synchronizes over the whole chain, while the larger frequency is limited to the Np (ω2 ) sites closest to i = N. In this situation, one finds that permanent synchronization domains for


157

ω1 and ω2 are established, with a domain wall, that evolves irregularly in space and time (Figure 5.12(b)). If one increases N , the ω2 domain is always confined to the last Np (ω2 ) sites, independently of the total length of the chain as well as of the value of ω1 . For the case of Figure 5.12(b), one can further plot the instantaneous frequency for i = 1, i = 40 and i = 22. It can be observed that the site i = 22 located on the domain boundary locks alternatively to ω1 and ω2 . The locking periods are interrupted by defects. Finally, for Np (ω1 ), Np (ω2 ) < N, neither one of the two frequencies stabilizes a synchronized pattern over the whole chain. In this case, one observes alternation between synchronization patterns with frequencies ω1 and ω2 , with intervals of asynchrony filled with defects, as shown in Figure 5.12(c). The duration of the synchrony and asynchrony intervals is irregular. This competitive behavior persists in time. An explanation for this behavior can be offered with reference to Figure 5.10(b), that illustrates the transient state for a wave with penetration depth Np < N . One, indeed, notices that the wave expands initially to the whole system but such a transient state is eventually broken by a defect, yielding a stationary regime in which the synchronization domain includes only the first Np sites. When two of these waves compete, the respective transient states alternates. As a consequence, the competition between the two frequencies has here a cooperative effect, insofar as it enhances the ability of each single entrainment process to reach global synchronization over finite time slots. In this way, Ref. [191] offered the description of a very intriguing scenario of competition between different synchronization domains, involving a series of interesting nontrivial states in which domains of synchronization can coexist separated by irregular domain walls.

5.6. Synchronization in continuous extended systems The natural continuation of the previous studies was to investigate synchronization between coupled spatially extended fields, i.e., systems modelled by partial differential equations. In fact, these studies were initiated in Refs. [195–200]. In the following, we will focus on the particular case of two fields obeying one dimensional Complex Ginzburg–Landau Equations (CGL), and will discuss the case of coupled identical CGL [201,202] and nonidentical CGL, as, in this latter situation, a direct comparison is possible between bidirectional coupling [202, 203], unidirectional coupling [204] and external forcing [205]. Though being a specific model system, CGL is a very generic equation, as it describes the universal pattern forming features close to the emergence of a spatially extended Hopf bifurcation [206]. As so, the validity of CGL has been proved in


158

the description and modelling of many different situations in laser physics [207], fluid dynamics [208], chemical turbulence [209], bluff body wakes [210], etc. We start our discussion with the case of a bidirectional symmetric coupling between two nonidentical one dimensional fields, each one of them ruled by the CGL. The system under study is A˙ 1,2 = A1,2 + (1 + iα1,2 )∂x2 A1,2 − (1 + iβ1,2 )|A1,2 |2 A1,2 + ε(x)(A2,1 − A1,2 ).

(5.20)

Here A1,2 (x, t) ≡ ρ1,2 (x, t)e(iψ1,2 (x,t)) are two complex fields of amplitudes ρ1,2 and phases ψ1,2 ; ∂x2 A1,2 stands for the second derivative of A1,2 with respect to the space variable 0 x L, L is the system extension, dot denotes a temporal derivative, α1,2 , β1,2 are suitable real control parameters, ε(x) is a space distributed coupling factor, and for the time being we will explicitly consider periodic boundary conditions. The synchronization of space-time chaotic states generated by equations (5.20) has been the subject of a series of recent papers [202,203]. Precisely, in Ref. [202], it was reported that the synchronization of two identical CGL (α1 = α2 , β1 = β2 ) can occur as a result of a coupling in a finite number Nc of controllers, i.e., with a coupling function given by: ε(x) =

Nc

εδ(x − xi ),

(5.21)

i=1

where xi indicates the position where the ith coupling acts, and therefore it is referred to as the position of the ith controller. Ref. [202] made use of an adaptive recognition [211] and control [212–215] method, that was successfully applied also to several other situations, such as recognition of deterministic dynamics from biophysical signals [216], control of infinite dimensional chaotic dynamics [217], targeting of chaos [218], secure communication processes [219], and filtering of noise using wavelet techniques [220]. It is important to notice that equation (5.21) indicates that the coupling acts only in discrete points of the mesh, which have been selected to be equally separated in space (xi − xi−1 = ξ ). To be rigorous, one should observe that perturbing equation (5.20) with a coupling of the kind of (5.21) would not give any effect: δ perturbations in space are not able to modify a fully developed partial differential equation. This is the reason why in equation (5.21) δ is not a δ-Dirac function, but it indicates that the coupling at position xi is extended over all the space extension of the ith controller, which usually corresponds to a single mesh point.


159

Notice that this is, indeed, the case in practical applications, where the controllers have in general a spatial size, and they’re never able to act on the system in a point-like region. From the other side, it has to be mentioned that locally space extended controllers have been also considered, specifically with the aim of studying how the extension of the controllers influences synchronization [221,222]. As an alternative choice, one can fix once forever the mesh precision, and then select accordingly the space extension of the controllers. This latter strategy (used extensively in Refs. [202,203]) will be summarized in the following. A first result obtained in Ref. [202] was that already a finite number of controllers is sufficient to warrant the complete synchronization of two identical systems. In the same Ref. [202] it was pointed out that the minimal number of controllers needed to robustly warrant synchronization between the two fields corresponds to a equi-spaced positioning function for the controllers in which one places a controller each approximately two correlation lengths (ξ 2ξc ) of the system. Successive studies were devoted to characterize synchronization of two fields coming from different dynamics (i.e., it selected α1 = α2 , β1 = β2 ), in the case in which the coupling function was extended over all the N mesh points (ε(x) ≡ ε). These studies have shown that a transition takes place for large parameter mismatches from no synchronization to complete synchronization, mediated by a state similar to phase synchronization. This intermediate state has been fully characterized in a recent work [204] for the case of a unidirectional coupling. Now, a question emerges on whether synchronization between nonidentical systems can be achieved with a discrete coupling. We now show that this is in fact the case in the limit ξ ξc provided that the coupling strength ε increases integrally as the number of controllers decreases. In order to describe correctly the situation, one has to start by discussing the main features of a single CGL. Different chaotic regimes can be identified in equations (5.20) in different regions of the parameter space (α, β) [223,224], depending on the stability properties of the plane wave solutions Aq = 1 − q 2 ei(qx+ωt) (−1 q 1, where q is the wave-number in the Fourier space, and ω = −β − (α − β)q 2 ). In particular, for αβ > −1 there exists a critical value 1 + αβ , qc = 2(1 + β 2 ) + 1 + αβ such that all plane waves in the range −qc q qc are linearly stable.

160


Such plane waves become unstable outside this range through the so-called Eckhaus instability [225]. Now, it is easy to realize that qc vanishes as the product αβ approaches −1. As a consequence, all plane waves will become unstable when crossing from below the so-called Benjamin–Feir line αβ = −1 in the parameter space. Above this line, three different turbulent states have been identified [223,224], namely phase turbulence (PT), amplitude turbulence (AT) or defect turbulence, and bi-chaos. In particular PT and AT have received a special attention in the scientific community [226–230]. PT is a particular turbulent state characterized by a chaotic evolution of the phase, whereas the amplitude changes smoothly, and it is always bounded away from zero. On the contrary, in AT the amplitude dynamics becomes chaotic. The large amplitude oscillations of the field can occasionally and locally cause the occurrence of a space-time defect in the point where the amplitude is vanishing. This implies that choosing in (5.20) a sufficiently large parameter mismatch between the equations governing the fields A1,2 is tantamount to selecting the uncoupled evolutions of the two fields to be in AT and PT respectively (for the time being α1 = α2 = 2.1, β1 = −1.2 and β2 = −0.83). The correlation length in the AT case (and for a spatial extension L = 256) is ξc = 5.38 [231]. In these conditions, Ref. [231] switched on the coupling term on a set of equi-spaced controllers, and studied the synchronization features emerging in the evolution of the two fields, as a function of both the coupling coefficient ε and the controllers number Nc . As a reference point, Ref. [231] described first the case of a distributed coupling coefficient (i.e., a coupling coefficient that is active for all mesh points, Nc = N ) for L = 256 and N = 2048. Precisely, in the long trial simulations performed by Ref. [231], the two systems A1,2 were left uncoupled for a time sufficient to extinguish the initial transient and to be in a chaotic AT (for A1 ) and PT state (for A2 ). After such an initial step, the coupling was switched on. Figure 5.13 shows the patterns arising from the space-time representations of ρ1 (a,c,e,g) and ρ2 (b,d,f,h) for ε = 0.05 (a,b), ε = 0.14 (c,d), ε = 0.2 (e,f), ε = 2 (g,h). One can easily see that, at small coupling strengths, the two systems do not evolve synchronously (a,b). At intermediate coupling (Figure 5.13(c,d)), the two systems enter both in a AT state and space-time defects (ρ2 = 0) are created in the system A2 . As the coupling further increases (Figure 5.13(e,f)), the two system return both in a synchronized PT state, and defects are no longer present. If one further increases the coupling (Figure 5.13(g,h)), the two states hold in a completely


161

Figure 5.13. Case: Nc = N . Space (horizontal) – time (vertical) plots of the moduli ρ1 (panels a,c,e,g) and ρ2 (panels b,d,f,h). α1 = α2 = 2.1, β1 = −1.2, β2 = −0.83. Time increases downwards from 500 to 1500 (u.t.). Note that the two systems were prepared in two independent chaotic states (amplitude turbulence for A1 and phase turbulence for A2 ). Panels (a) and (b) correspond to ε = 0.05; panels (c) and (d) to ε = 0.14; panels (e) and (f) to ε = 0.2; panels (g) and (h) to ε = 2. Reprinted with permission from Ref. [231]. © 2000 The American Physical Society


162

synchronized configuration, but creation-annihilation of defects in both systems takes place again. In order to quantitatively describe the above scenario, Ref. [231] made use of some indicators, such as the number of defects Ndef that are present in the system. Theoretically, a defect is a point (x, t) for which ρ(x, t) → 0, i.e., defects are intersections of the 0-level curves in the (x, t) plane of the real and imaginary parts of A1,2 (x, t). In practice, because of the finite size of the mesh, we count as defects at time t those points xi where the ρ(xi , t) is smaller than 2.5 × 10−2 and that are furthermore local minima for the function ρ(x, t). In Figure 5.14(a), a statistics is made of Ndef as a function of ε. The inset reports the situation for small coupling strength and corresponds to the first three cases of Figure 5.13. The number of defects of A1 is decreasing to zero when the coupling ε is increased. As for A2 , its defect number versus ε is not evolving monotonically, but it increases to a maximum value (for ε 0.12 where the number of defects is equal to the number of defects for A1 ) and then decreases to reach zero for ε 0.16. However, a further increase of the coupling generates an increasing of the defect number (e.g., for ε = 2, Ndef = 79). Two others indicators used in Ref. [231] are reported in Figure 5.14(b), namely the average (in space and time) of the modulus difference (solid line) =
x,t ρ 1 + ρ2

(5.22)

and the same average for the phase difference (dashed line) =
x,t . |ψ1 | + |ψ2 |

(5.23)

The inset again reports the situation at small coupling values. The modulus difference decreases much faster than the phase difference, as the coupling increases, and therefore one can infer that the defects first synchronize before having a complete synchronization state. Finally, in Figure 5.14(c,d) two other indicators are shown, namely SYG(t, ε) = x ,

(5.24)

SYG(ε) = <SYG(t, ε)>t ,

(5.25)

and

which are hybrid indicators between the phase and modulus difference indicators. Precisely, Figure 5.14(c) shows SYG(ε). This indicator has the property of not being a monotonous function of ε.


163

Figure 5.14. Case: Nc = N . Different indicators of synchronization. In panel (a), the total number of defects is reported as a function of the coupling strength ε for A1 (solid line) and A2 (dashed line). The inset reports the zoom at small ε values. In panel (b), the modulus (solid line) and phase (dashed line) indicators are plotted as functions of ε (see text for definition). The modulus indicator has been multiplied by a factor 7 in order to get the same vertical scale for the two indicators. The inset reports the zoom at small ε values. In panel (c) the SYG indicator (see text for definition) is reported vs. ε. The inset reports, again, the zoom at small ε values. Finally, in panel (d) the temporal evolution of the SYG indicator (in arbitrary units) in sketched for a fixed coupling strength ε = 2. The inset represents a zoom from t = 600 to t = 800. Parameters are the same as in the caption of Figure 5.13. Reprinted with permission from Ref. [231]. © 2000 The American Physical Society

In particular, when the number of defect vanishes, a sudden increase in the corresponding value of the SYG indicator is observed. Figure 5.14(d) reports the temporal evolution of SYG(t, ε) at large coupling (ε = 2), indicating that the synchronization is not a stable process, and that some kind of intermittency phenomena occur. After having performed the full analysis for the case of distributed coupling, Ref. [231] moved to describe what is happening in the case of a discrete coupling, and fixed Nc = N/5.

164


Figure 5.15 shows the space-time plots of ρ1 (a,c,e,g) and ρ2 (b,d,f,h). One immediately notices that in Figure 5.15 ε = 0.25 (a,b), ε = 0.70 (c,d), ε = 1 (e,f), and ε = 10 (g,h) which are exactly five times the coupling strengths used by Ref. [231] in Figure 5.13 for Nc = N . Furthermore, by comparison with Figure 5.13, it is nearly impossible to distinguish between the two scenarios of synchronization stages. Figure 5.16 reports the behavior of the above indicators in the case Nc = N/5, showing the same succession of events as for discrete synchronization, when the coupling strength is multiplied by the factor Nc /N . As an ultimate test to show up the validity and limitations of this integral behavior was performed in Ref. [231], consisting in defining ε∗ as the minimum coupling strength for which the number of defects in A1 and A2 vanishes, and by reporting ε ∗ vs. ξ/xic . The outcome is reported in Figure 5.17, where the integral behavior is witnessed by the fact that the results align quasi perfectly on a straight line for ξ/ξc 1/3. The general conclusion of Ref. [231] was therefore that a discrete set of equi-spaced controllers is already sufficient for determining the emergence of a complete synchronization phenomenon for coupled nonidentical space extended fields, providing that, in the limit of more than three controllers per correlation length, the value of the coupling increases integrally with the controller space separation.

5.7. Asymmetric coupling effects The next step is to fully discuss the effects of asymmetries in the coupling of space-extended continuous fields. To this purpose, we will follow the approach of Ref. [232], and consider the explicit case of two asymmetrically coupled CGL described by: A˙ 1,2 = A1,2 + (1 + iα)∂x2 A1,2 − (1 + iβ1,2 )|A1,2 |2 A1,2 c + (1 ∓ θ )(A2,1 − A1,2 ). (5.26) 2 Here again, A1,2 (x, t) = ρ1,2 (x, t)eiφ1,2 (x, t) are two complex fields (of amplitudes ρ1,2 (x, t) and phases φ1,2 (x, t)), dots denote temporal derivatives, ∂x2 stays for the second derivative with respect to the space variable 0 x L, L is the system extension, α and β1,2 are suitable real parameters, c represents the coupling strength. The important parameter of equations (5.26) is θ , that is accounting for asymmetries in the coupling. Indeed, the case θ = 0 recovers the bidirectional symmetric coupling configuration described in the previous section, whereas the case θ = 1 (θ = −1) describe the unidirectional master slave scheme, with the field A1 (A2 ) driving the response of A2 (A1 ).


165

Figure 5.15. Case: Nc = N/5. Space (horizontal) – time (vertical) plots of the moduli ρ1 (panels a,c,e,g) and ρ2 (panels b,d,f,h). α1 = α2 = 2.1, β1 = −1.2, β2 = −0.83. Time increases downwards from 500 to 1500 (u.t.). Same general stipulations as in the caption of Figure 5.13. Panels (a) and (b) correspond to ε = 0.25; panels (c) and (d) to ε = 0.70; panels (e) and (f) to ε = 1; and panels (g) and (h) to ε = 10. Reprinted with permission from Ref. [231]. © 2000 The American Physical Society

166


Figure 5.16. Case: Nc = N/5. Same indicators of synchronization as in Figure 5.14. Panel (a): Total number of defects versus the coupling strength ε for A1 (solid line) and A2 (dashed line). The inset reports the zoom at small ε values. Panel (b): The modulus (solid line) and phase (dashed line) indicators vs. ε. The modulus indicator has been multiplied by a factor 7 in order to get the same vertical scale. The inset reports the zoom at small ε values. Panel (c): SYG indicator vs. ε. The inset reports the zoom at small ε values. Panel (d): SYG indicator vs. time (in arbitrary units) for a fixed coupling strength ε = 2. The inset is a zoom from t = 1100 to t = 1300. In panels (a,b,c) ε(5) means that we take one controller each five mesh points. Notice that the corresponding transitions occur for coupling strengths five times larger than in Figure 5.14. Reprinted with permission from Ref. [231]. © 2000 The American Physical Society

On the other hand, θ is a real parameter that can, in principle, assume any value in the range [−1, 1]. In this way one can investigate any possible coupling scheme intermediate between the unidirectional and the bidirectional one, and see how such a parameter affects the synchronization properties of system (5.26). In order to properly characterize the synchronization properties of the coupled fields, Ref. [232] made use of the following indicators [204]. In order to reveal 1:1 phase synchronization, instead of using the phases φ1,2 (x, t) ∈ [0, 2π], Ref. [232] made reference to the unfolding of the phases to the real axis (φ1,2 (x, t) ∈ R).


167

Figure 5.17. ε∗ (see text for definition) as a function of the ratio ξ/ξc . The straight line aligning the results witnesses the existence of an integral behavior for the discrete synchronization in the limit of at least three controllers per spatial correlation length. Reprinted with permission from Ref. [231]. © 2000 The American Physical Society

In this framework, phase synchronization is established if the following condition is satisfied: φ = max φ1 (x, t) − φ2 (x, t) < K, (5.27) x∈L, T t∈R

where T denotes a transient time, and K is a suitable real number. Notice that the condition (5.27) implies that the maximum relative phase difference remains bounded for all times. Furthermore, in (5.27), Ref. [232] introduced a transient time T , in order to get rid of all transient effects (in the present case T = 4000), and to monitor exclusively the settings of asymptotic synchronous states. A second measure of frequency locking used in Ref. [232] is the monitoring of the mean frequency mismatch. The mean frequency of each field is given by x , t where x denotes spatial average. 1:1 frequency synchronization occurs when Ω1,2 = lim

t→∞

Ω = Ω1 − Ω2 = 0.

(5.28)

It is important to notice that condition (5.28) represents a weaker form of phase locking, because a frequency locking does not prevent the evolution of φ from being affected by 2π phase slips over secular time scales.

168


As for complete synchronization states, Ref. [232] used the measure of Pearson’s coefficient γ , which retains the degree of cross correlation between the moduli ρ1,2 (x, t) (once again after the transient time T ) (ρ1 − ρ1 )(ρ2 − ρ2 ) , γ = (ρ1 − ρ1 )2 (ρ2 − ρ2 )2

(5.29)

where denotes a full space–time average. Precisely, when γ 0 the two fields are linearly uncorrelated; γ = 1 marks complete correlation and γ = −1 indicates that the fields are negatively correlated. As in the previous section, another indicator for the disorder in the system is the number of phase defects N . It is well known that N is an extensive quantity of both time and space, and therefore it is sometimes convenient to refer to the defect density nD , that is calculated as the defect number N per unit time and unit space. In order to describe the effects of asymmetry in the coupling in system (5.26), Ref. [232] selected α = 2, β1 = −0.7, and β2 = −1.05, which corresponds to set the uncoupled evolution of A1 (A2 ) in PT (AT). In all cases, the different synchronization indicators were evaluated over a time t = T + 6000. Figure 5.18 reports the Pearson’s coefficient values in the parameter space (c, θ ) (a) as well as two cuts of the surface at c = 0.25 and c = 0.6 (b). The results indicate that the threshold for the appearance of a complete synchronization state depends crucially on the asymmetry θ in the coupling. The next step is to discuss how asymmetry influences the selection of the final synchronized state. Looking at Figure 5.19, one realizes that the CS state (γ 1) occurs in PT (no defects in both fields), for nearly all values of θ . This confirms the results already shown in the previous section for θ = 0 and indicates that the preferred state for complete synchronization is in the PT regime, except for θ very close to −1, where the AT system is driving the “slaved” PT system. Even more interestingly, the final CS state is reached after an intermediate (γ < 1) AT state, where both systems displays phase defects, and the number of defects in the field A1 (that was originally set in PT) shows a kind of divergence for coupling strength close but slightly below the threshold for CS (Figure 5.19(b)). Namely, simulations performed at c = 0.38 and θ = −1 with a system of larger extent L = 1000 show that the PT system produces approximately 19 times more defects than the AT system within the intermediate state. This is shown in Figure 5.20 where the space–time plots of both ρ1 (a) and ρ2 (b) are reported, and a much larger number of defects characterize the behavior of the field originally set in PT. Finally, Ref. [232] investigated how asymmetric couplings influence the settings of phase and frequency synchronization states. Figure 5.21 reports the indi-


169

Figure 5.18. Upper plot: Pearson’s coefficient γ (see text for definition) vs. the parameter space (c, θ). Other parameters are α = 2, β1 = −0.7 and β2 = −1.05. Lower plot: Pearson’s coefficient vs. the coupling asymmetry parameter θ , for c = 0.25 (dashed line) and c = 0.6 (solid line).

cators for phase synchronization (a) and frequency synchronization (b), as well as a plot of the maximum phase difference vs. time (c) for the parameters c = 0.6 and θ = −0.9. Once again, one notices that the thresholds for the setting of phase and frequency synchronized states depend on the asymmetry, and both synchronization features are enhanced for θ ⇒ 1 (see Figure 5.21(a,b)).

170


Figure 5.19. Total number of defects N generated in the system A2 (panel a) and A1 (panel b) in the parameter plane (c, θ). Other parameters as in the caption of Figure 5.18. The density of space–time defects nD can be obtained here by normalizing N to 6000x100 (time interval x space interval).

In the opposite limit (θ ⇒ −1), by comparing Figure 5.21(a) with Figure 5.18, one realizes that phase synchronization is set for coupling strengths for which a complete synchronized state has already emerged in the system, thus the range for the existence of a phase synchronization regime is here shrunk. At variance, frequency synchronization persists also for θ −1 (see Figure 5.21(b)). More insight on the limits for phase synchronization states can be extracted from Figure 5.21(c), where the maximum phase difference is reported vs. time for c = 0.6 and θ = −0.9 (parameters for which frequency synchronization is observed). There, one can see that the system shows rather long epochs of phase locked states, interrupted by sudden 2π phase slips, that occur over secular time scales.


171

Figure 5.20. Space–time plot of ρ1 (x, t) (left panel) and ρ2 (x, t) (right panel) for c = 0.38 and θ = −1. Time is increasing upwards. Other parameters as in the caption of Figure 5.18. Notice that here the system originally evolving in phase turbulence is characterized by the presence of a number of defects approximately 19 times larger than that characterizing the system originally evolving in amplitude turbulence.

As a consequence, the time averaged frequency difference vanishes, whereas the phase difference is unbounded. Another effect of asymmetry is the transition from normal to anomalous frequency synchronization states, as it can be seen in Figure 5.21(b). “Anomalous” phase synchronization was recently introduced for coupled confined systems, for indicating a situation in which increasing the coupling initially leads to a degradation of frequency locking. Here, for almost all θ (except for θ ≈ 1), an increase in c yields initially a higher frequency difference. After reaching a maximum, Ω eventually vanishes as c approaches the asymmetry dependent threshold for frequency synchronization. When θ approaches 1, the final synchronized state is reached by means of a smooth transition in Ω, because the coupling scheme approaches the unidi-

172


Figure 5.21. Panel (a): Maximum value of |φ(t)| (in logarithmic scale) vs. the parameter space (c, θ). Panel (b): Absolute value of Ω vs. the parameter space (c, θ). Panel (c): |φ| vs. time for c = 0.6 and θ = −0.9. Notice that the function is increasing by finite jumps (2π phase slips) giving rise to a unbounded stair-like behavior. Other parameters are the same as in the caption of Figure 5.18.


173

rectional configuration with the phase turbulent subsystem forcing the amplitude turbulent one.

5.8. Defect enhanced anomaly in asymmetrically coupled spatially extended systems We now pay a special attention to describe in more details the anomalous frequency synchronization. In particular, we will follow the approach of Ref. [233], that analytically established and numerically showed that: • anomalous frequency synchronization (AFS) is a generic phenomenon, not limited to the case of coupled low-dimensional systems, but occurring also for space extended systems, • the transition to anomalous behaviors is crucially dependent on asymmetries in the coupling configuration, and • the presence of phase defects in spatially extended chaotic oscillators has the role of enhancing the anomaly in frequency synchronization with respect to the case of merely time chaotic oscillators. Ref. [233] started once again by considering a pair of asymmetrically coupled Complex Ginzburg–Landau equations ruled by equations (5.26), and concentrated on the case in which both fields are initially (for c = 0) set in a regime of phase turbulence (PT), i.e., the parameters in equation (5.26) are α = 2, β1 = −0.75 and β2 = −0.9. The condition for 1:1 frequency synchronization is again the vanishing of the mean frequency mismatch equations (5.28). Figure 5.22 reports Ω in the parameter space (c, θ) and indicates that the transition to a frequency locked state (Ω = 0) can occur in a regular (Ω is a monotonically decreasing function of c) or in an anomalous way (Ω increases initially with c), depending upon the level of asymmetry in the coupling configuration. The arrow in Figure 5.22 indicates the critical value θcr ≈ −0.09 and marks the numerically found transition point between the two frequency synchronization behaviors. As the two fields are initially set in the PT regime, Ref. [233] made use of the tools of asymptotic analysis in order to reduce the description to a pair of coupled Kuramoto–Sivashinsky (KS) equations. We here summarize this latter approach for a single CGLE equation, and we will later consider the problem of adding the coupling term. The equation is: A˙ = A + (1 + iα)∂xx A − (1 + iβ)|A|2 A.

(5.30)

In the PT regime, the dynamics of the complex field A(x, t) can be reduced to the dynamics of the real phase field φ(x, t), being the amplitude field ρ(x, t) slaved to the dynamics of the phase.

174


Figure 5.22. Frequency mismatch |Ω| vs. the parameter space (c, θ) for equations (5.26). The arrow located at θcr ≈ −0.09 discriminates between regular (θcr < θ 1) and anomalous (−1 θ < θcr ) frequency synchronization. Reprinted with permission from Ref. [233]. © 2005 The American Physical Society

The family equations that describe the dynamics of the phase equations are called Kuramoto–Sivashinsky equations. Such equations have been found to properly describe chemical reactions (e.g., Belousov–Zhabotinsky reaction) as well as flame propagation in the case of mild combustion. The equation for the phase dynamics of the single CGLE has been derived by Sakaguchi [234] and reads as: 2 φ˙ = t1 φxx + t2 φx2 + t3 φxxxx + t4 φx φxxx + t5 φxx + t6 φx2 φxx

−α 2 (1

(5.31)

where t1 = 1 + αβ, t2 = β − α, t3 = + t4 = −2α(1 + β 2 ), 2 2 t5 = −α(1 + β ) and t6 = −2(1 + β ). Notice that equation (5.31) is obtained by doing an asymptotic expansion of equation (5.30) in powers of ∂x , using the degree of spatial modulation of the phase as the smallness parameter for the expansion. As initial condition for equation (5.31), Ref. [233] selected a Gaussian noise with zero mean and standard deviation σ = 10−4 . In equation (5.31), after some transient, the phase φ is drifting linearly with time (x ≈ st + b, where s is the slope of the linear drift evaluated by performing a linear fit). The validity of the KS reduction model can be checked by comparing the average frequency obtained from the full CGLE (equation (5.30)) and the frequency β 2 )/2,


175

Figure 5.23. Panel (a): Time evolution of x from equation (5.30) with α = 2 and β = −0.75. Panel (b): Probability distribution function of the data shown in (a) (dotted line) and its best fit by a Gaussian PDF (solid line). Panel (c): Time auto-correlation function of the signal shown in (a). Reprinted with permission from Ref. [233]. © 2005 The American Physical Society

estimate given by the KS model ω = −β + s, and Ref. [233] found an excellent agreement in the whole PT regime (−0.9 < β < −0.7). In fact, as one is here interested in a first order perturbation theory, one can limit the analysis to the first 3 terms in the right-hand-side of equation (5.31). The advantage of the 3 terms reduction of equation (5.31) is that it is now straightforward to perform a spatial average of such reduced equation, that leads to: ˙ x = t2 x .

(5.32)

Equation (5.32) is a very simple relationship for the correction to the frequency ˙ x ). In Figure 5.23(a), the time evolution of the term (from (ω = −β + now on referred to as T2) is displayed, as it is taken from the simulation of the full CGLE. The time evolution of T2 is clearly chaotic and its probability distribution function (PDF) can be conveniently fitted by a Gaussian (as shown in Figure 5.23(b)). Furthermore, the time correlation function for T2 is reported in Figure 5.23(c). There, by assuming an exponential decay, Ref. [233] obtained an estimate for the correlation time of τ = 51.2.


176

Such statistical properties made it possible to further reduce the analysis to a pair of time dependent coupled oscillators (GL equations) subjected to a colored Gaussian noise with proper mean, fluctuation and correlation features. Taking back into account the coupling term, using proper noise terms and assuming small parameter mismatches, the reduced GL model [235] for the phases φ1 (t), φ2 (t) of the chaotic oscillators becomes:

φ˙ 1 = −β1 − c1 β1 (cos ϕ − 1) − sin ϕ + η1 ,

(5.33) φ˙ 2 = −β2 − c2 β2 (cos ϕ − 1) + sin ϕ + η2 , where ϕ = φ2 − φ1 represents the phase difference between the two oscillators, c1 = c(1 − θ )/2, c2 = c(1 + θ )/2, and η1 , η2 are the two colored Gaussian noise terms specified above. Equations (5.33) are a set of stochastic differential equations (SDE) where the noise terms η1 , η2 have been surrogated from the full CGLE with the corresponding parameters α, β and c = 0 (uncoupled). Before proceeding with numerical integration, Ref. [233] made some analytical studies of equations (5.33), following a similar approach that was proposed in Ref. [236]. Namely, by neglecting the noise terms and subtracting equations (5.33), one is able to write an equation for φ˙ in a closed form. Then frequency synchronization is studied by calculating the frequency detuning −1

(Ω)

1 = 2π

2π 0

yielding |Ω| =

dϕ , ϕ˙

2 + cB [θ B − B ] + c2 , B− − + −

(5.34)

where B+ (B− ) stays for β1 + β2 (β1 − β2 ). The interest of expression (5.34) is that one can analytically estimate the transition point between normal (the slope of the detuning at zero coupling is negative) and anomalous (the slope of the detuning at zero coupling is positive) frequency synchronization. It is therefore straightforward to calculate the value of θ for this transition as: B− β1 − β2 = . θcr = (5.35) B+ β1 + β2 For the particular case treated in Ref. [233] (β1 = −0.75 and β2 = −0.9), one has θcr = −1/11 ≈ −0.09, in perfect agreement with what found numerically for the full CGLE model and reported in Figure 5.22. Finally, Ref. [233] compared the numerical integration of the SDE (5.33) and of the full CGLE (5.26). Figure 5.24 reports the frequencies Ω1 , Ω2 vs. the coupling


177

Figure 5.24. Mean frequencies Ω1,2 calculated for the full CGLE (5.26) (solid line) and the effective SDE (5.33) model (dashed line). Panel (a) corresponds to θ = 0.88, and describes the situation occurring during regular frequency synchronization. Panel (b) corresponds to θ = −0.88, where anomalous frequency synchronization occurs. In both cases β1 = −0.75, β2 = −0.9. Reprinted with permission from Ref. [233]. © 2005 The American Physical Society

strength c for two asymmetric coupling configurations. Namely, Figure 5.24(a) (Figure 5.24(b)) refers to the case of regular frequency synchronization at θ = 0.88 (of anomalous frequency synchronization at θ = −0.88). The agreement between the SDE (5.33) and the full CGLE (5.26) is very good at low coupling strengths. However, for larger values of c, an increasing difference is observed between the two cases, which is especially pronounced in the case of anomalous frequency synchronization. In particular, it comes out that the full CGLE shows an enhancement of the anomaly with respect to the SDE (5.33). This is entirely due to the presence of phase defects for the frequency miseff match observed in Figure 5.24. Calling Ω1 (Ω1 ) the mean frequency of system 1 as calculated with reference to the SDE model (the full CGLE model), in Figure 5.25 it is shown that for θ = −0.88, the 1:1 correlation between the frequency eff mismatch Ω1 − Ω1 and the number of defects appearing in system 1 is indeed remarkable, indicating that a simple correction of the frequency proportional to the defect numbers is enough to produce an excellent agreement between the SDE and the full CGLE models for the whole range of c.

178


eff

Figure 5.25. Number of defects in system 1 (solid line) and frequency mismatch Ω1 − Ω1 (dashed line, see text for definition) vs. c for θ = −0.88 during anomalous frequency synchronization. To make visible the comparison, frequencies have been multiplied by 2.5 104 . It is remarkable to observe that a simple correction of the frequency proportional to the defect numbers is sufficient to produce agreement between the stochastic differential equation model and the full model of Complex Ginzburg Landau Equation for the whole range of c. Reprinted with permission from Ref. [233]. © 2005 The American Physical Society

The important conclusion of Ref. [233] was to show that anomalous effects in frequency synchronization are present also in coupled fields, where they are actually enhanced by the presence of spatial phase defects, i.e., structures that are intimately related to the space-extended nature of the coupled fields.

5.9. Convective instabilities of synchronization in space distributed and extended systems To conclude this chapter dedicated to synchronization in space-extended fields, a final, crucial, point to be discussed concerns the fact that, in such systems, other mechanisms, different from transversal stability, can lead to the destruction of the synchronous motion. As far as confined chaotic systems are concerned, indeed, the stability of the synchronization motion can be fully assessed by the study of the transversal Lyapunov exponent or the Lyapunov function, as we have discussed largely in Chapter 2.

5.9. Convective instabilities of synchronization in space distributed and extended systems 179

On the opposite, for space extended systems, absolutely stable synchronization manifolds can be affected by other types of space-time instabilities leading to desynchronization. This point was manifested recently in Ref. [237], where it was shown that, for chains of unidirectionally coupled identical chaotic oscillators, an absolutely stable anticipating synchronization manifold (ASM) can be nevertheless convectively unstable and thus looses its synchronization properties for any small noisy perturbation. Let us, here, remind that anticipating synchronization [238] was initially proposed as a property of identical chaotic oscillators, unidirectionally coupled in a driver-response configuration by means of a memory element in the coupling factor (represented by a short time delay τ ). In anticipated synchronization, the trajectories of the two systems converge toward an absolutely stable manifold, characterized by the fact that the state of the response system anticipates the one of the driver by the same amount of time τ . Such a feature was proven to be a stable property of pairs of chaotic systems, and later extended to open chains of identical systems [238]. To make an explicit example, we follow the approach of Ref. [237], and consider an open chain of N unidirectionally coupled identical Rössler oscillators [34], given by

r˙ i = f(ri ) + ε(1 − δ1i ) ri−1 − ri (t − τ ) . (5.36) Here, dots denote temporal derivatives, ri ≡ (xi , yi , zi ) is the vector field of the ith driven oscillator (i = 1, . . . , N), is the coupling strength, τ is the delay time in the coupling factor, δij is the Kronecker δ function, and f(r) is a vector field

f(r) = −y − z, x + ay, b + z(x − c) (5.37) responsible for generating the locally chaotic dynamics. Ref. [237] set a = 0.15, b = 0.2, c = 10, N = 100, and studied the evolution of system (5.36) for different values of τ and . It has to be remarked that, in order to evolve the dynamics of system (5.36), one has to start from a set of random initial conditions ri (0) covering all the interval [0, −τ ] for each oscillator. In particular, the interest is to study the stability properties of the anticipating synchronization manifold, so that one has to recur to linear stability analysis. This is done by passing from the vectors {ri (t)} to the representation r1 (t), ri ≡ ri−1 (t) − ri (t − τ ) , with i > 1. In this framework, the synchronized state is ri = 0, while the linearization of the equations for r1 accounts for the Lyapunov exponents of the single Rössler oscillator.

180


Ref. [237] considered the dynamics of an infinitesimal perturbation ρi = (ui , vi , wi ) of the differences ri described by u˙ i = −vi − wi + (1 − δ1i )ui−1 − ui (t − τ ), v˙i = ui + azi + (1 − δ1i )vi−1 − vi (t − τ ), w˙ i = (xi − c)wi + zi ui + (1 − δ1i )wi−1 − wi (t − τ ).

(5.38)

The growth rates of ρi give information on the evolution of perturbations transversal to the anticipating synchronization manifold (represented here by the fixed point ρ = 0), and a necessary condition for the synchronized regime to be stable is that the growth rate λ⊥ of ρ0 is negative. Consistently with what observed in Ref. [238] for a pair of coupled systems, Ref. [237] pointed out that λ⊥ can be, indeed, negative in a suitable parameter range, marking absolute stability for the anticipating synchronization manifold. However, this fact could lead one to conclude that, once selecting the parameters in the absolute stability range, arbitrarily long anticipation times can be obtained by just coupling a sufficiently large number N of oscillators, since the ith oscillator anticipates its driver by a time τ , and its dynamics is therefore expected to collapse onto a manifold wherein ri (t) = r1 t + (i − 1)τ . The above statement would be in stringent contrast with the chaotic nature of the uncoupled systems, as it is clear that no trustable anticipations can be made for a chaotic system for time scales larger than the inverse of the maximum Lyapunov exponent. This apparent paradox, in fact, is easily reconciliated to the fact that the system under study is intrinsically space extended, and therefore one has to consider here that absolute stability alone cannot be taken as a sufficient condition for the settings of such a manifold. Indeed Ref. [237] showed clearly that this is not the case. If, system (5.36) is let evolve from random initial condition for N = 100, τ = 0.1 and ε = 4.1 (these parameters fall inside the region where λ⊥ is negative), a zero average δ-correlated Gaussian noise Dξ(t) of small amplitude D = 0.005 added to the variable x2 makes that the trajectory abandons the absolutely stable manifold δxi = 0, with an asymptotic (in time) size of the deviations that tend to grow exponentially with i. In order to clarify the de-synchronization process, Ref. [237] investigated the response of the system to a delta-like perturbation. This can be done by letting the system (5.36) evolve from random initial condition at t = 0 (with τ = 0.1 and ε = 4.1) until it reaches (within numerical accuracy) the synchronization manifold. At this stage, the evolution is restarted after perturbing x1 by a small amount η (while all other variables are left unchanged) and the convergence back to the


Figure 5.26. Time evolution of the ensemble averaged differences σi2 = u2i for i = 25, 50, 75 and 99 (the corresponding numbers are on the top of each curve). Each curve is obtained from an ensemble average over 10,000 perturbations, for τ = 0.1, ε = 4.1 and η = 5 × 10−3 . Solid (dotted) line reports the calculation in the normal (tangent) space. In all cases it is evident that the deviation from the synchronization manifold initially grows but asymptotically vanishes. Reprinted with permission from Ref. [237]. © 2004 The American Physical Society

anticipating manifold is studied by monitoring the single step anticipation error

2 σi2 = xi (t − τ ) − xi−1 (t) , where angular brackets denote an average over an ensemble of independent choices of the initial conditions. Practically, in the limit of small perturbations, instead of following two separate trajectories, it is sufficient to let a perturbation evolve in the tangent space: in this limit σi2 = u2i . The curves corresponding to different oscillators were reported by Ref. [237], and are plotted in Figure 5.26. They clearly indicates that the deviation from the manifold initially grows but eventually converges to 0 thus confirming its absolute stability. On the other hand, one notices that oscillators labelled by larger i-values are characterized by higher peaks. The observed phenomenology shares a lot of similarities with the phenomenon of convective instabilities in spatially extended systems, where a localized perturbation that is washed out when observed in the same point where it has been


182

applied, it instead grows in time when monitored in a suitable moving frame at a given constant velocity. And Ref. [237], indeed, numerically tested whether the evolution of an initially localized perturbation follows the same scaling behavior as the one due to convective instabilities in spatially extended systems. It is useful to remind that, in the context of one-dimensional lattices, the convective Lyapunov exponent is defined as [239] 1 |δ(i = vt, t)| ln (5.39) t δ(0, 0) where δ(i, t) denotes the perturbation amplitude in site i at time t and is initially localized in a finite region around the origin. This is tantamount to say that Λ(v) δ(i, t) exp Λ(v)t = exp (5.40) i v Λ(v) = lim

t→∞

for large enough |i| and t. Now, Λ(v) can be numerically estimated by comparing the perturbation amplitude at two different space-time positions P1 ≡ (i1 , t1 ), and P2 ≡ (i2 , t2 ), in the following way |δ(i2 , t2 )| v ln , (5.41) i2 − i1 |δ(i1 , t1 )| where v = i1 /t1 = i2 /t2 . This is because, in the limit in which both P1 and P2 are far enough from the origin, all multiplicative finite-size corrections would affect δ in the same way, and therefore they would disappear when the ratio is taken in equation (5.41). The numerical results illustrated in Ref. [237], and reported in Figure 5.27, confirm that the behavior of perturbations in the context of Rössler oscillators with delayed coupling is analogous to that of convectively unstable systems. The three curves obtained by comparing the pairs of oscillators (80, 60), (60, 40), and (40, 20) almost overlap, thus suggesting that the convective spectrum Λ(v) is a well defined quantity also in this context. The existence of a positive maximum of Λ(v) implies that perturbations travelling with a velocity v in between the two zeros of Λ(v) (approximately equal to 5 and 8) are amplified. Furthermore, the value of the maximum convective exponent can be independently checked by monitoring the values of the maxima σM of each σi versus their occurrence time. The best fit reported in the inset of Figure 5.27 corresponds to a growth rate of 0.202, in good agreement with the maximum of the convective spectrum. It is important to remark that the maximum Lyapunov exponent for the single Rössler oscillator is, for the parameters selected in Ref. [237], much smaller that the maximum convective exponent. This implies that these new convective instabilities are even stronger than the intrinsic instabilities of the uncoupled systems. Λ(v) =


Figure 5.27. Convective Lyapunov exponent Λ(v) vs. the propagation velocity v, computed by comparing the perturbation in different pairs of oscillators according to equation (5.41). Dot-dashed, dotted and solid lines correspond to the pairs (80, 60), (60, 40), and (40, 20), respectively. The maximum value of the exponent is marked by an arrow. The inset reports the maximum value of σi vs. its time of occurrence. The rate of an exponential best fit (dashed line in the inset) equals to 0.202 and fully agrees with the maximal convective Lyapunov exponent. Reprinted with permission from Ref. [237]. © 2004 The American Physical Society

The general conclusion of Ref. [237] evidences that absolute stability of the synchronization manifold is only a necessary conditions for the robustness of synchronization in coupled spatially extended systems, whereas the stability of synchronization strictly depends on the space-extended nature of the dynamics and needs to be assessed by taking into account additional sources of instability, such as convective growth of perturbations in moving frames.

Chapter 6

Complex Networks

Sempre caro mi fu quest’ermo colle e questa siepe, che da tanta parte dell’ultimo orizzonte il guardo esclude. Ma sedendo e mirando, interminati spazi di la’ da quella (...) e profondissima quiete io nel pensier mi fingo, ove per poco il cor non si spaura. (...) e in questa infinita’ s’annega il pensier mio, e il naufragar m’e’ dolce in questo mare. Giacomo Leopardi1 The last chapter of this book is devoted to give a rapid survey of synchronization phenomena in complex networks of dynamical elements, i.e., in the situation in which an ensemble (usually large) of chaotic systems is coupled by means of a wiring of connection that has particular topological properties, and it can even be time-dependent. This subject, indeed, has recently attracted a lot of attention in the scientific community, as many natural, technological and social systems find a suitable representation as networks made of a large number of highly interconnected units (this is the case, for instance, of electric power grids, the Internet, biological, neural and chemical networks, or networks of acquaintances, friendships or collaborations between individuals). 1 Always dear to me was this lonely hill

and this hedge, that from so much of the latest horizon my glance excludes. But sitting down and watching, endless spaces behind it (...) and the deepest stillness I figure out in my mind, and my heart is about to be scared. (...) and in such an infinity my thought drowns, and being shipwrecked is so agreeable in this sea. 185

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Chapter 6. Complex Networks

Furthermore, recent studies have revealed that such natural, technological and social systems are all characterized by similar topological properties: relatively small characteristic distances between any two nodes, high clustering properties, and fat tailed shapes in the distribution of connectivities. Actually, the subject of synchronization in complex networks would deserve itself to be conveniently summarized into a book, especially because there are several important questions related to the interplay between topological complexity in the wiring of connections and dynamical complexity of the elements constituting the nodes of the graph. In particular, as far as synchronization is considered, one has to clarify to which extent the properties of the emerging collective behavior depends on the topological structure of the underlying connections, or to which extent synchronization itself can contribute to reshape and change in time the topology characterizing the global connectivity of the network. In the following, we will limit to illustrate the main concepts and ideas that have been developed in the recent years, and that make synchronization phenomena in complex networks different from what already described in the other chapters of the present book. In particular, we will focus on reporting those studies on synchronization of networking units that tried to understand the main mechanisms through which the complexity in the overall topology and the local dynamical properties of the coupled units concur to give rise to a synchronous motion [240,241]. In other words, we will mostly refer to the main techniques that have been proposed for assessing the propensity for synchronization (synchronizability) of a given networked system. The chapter starts with a brief introduction to the main definitions and quantities that characterize a complex network, and then describes the main features of synchronous motion in such a kind of systems, especially in the view of selecting the optimal topology in the coupling configuration that provides enhancement of the synchronization features. The chapter ends with the summary of recent techniques that show how to conveniently employ synchronization properties of complex networks for gathering relevant information on the underlying topological structure of the graph.

6.1. Definitions and measures of complex networks 6.1.1. Unweighted graphs Many natural systems, such as coupled biological and chemical systems, neural networks, social interacting species, the Internet and the World Wide Web, can be conveniently modelled as networks composed by a large number of highly interconnected dynamical units.

6.1. Definitions and measures of complex networks

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A network can be represented formally as a graph. A undirected (directed) graph G = (N , L) consists of two sets N and L, such that N = ∅ and L is a set of unordered (ordered) pairs of elements of N . The elements of N ≡ {n1 , n2 , . . . , nN } are the nodes of the graph, while the elements of L ≡ {l1 , l2 , . . . , lK } are its links or edges. While each node of the graph is usually referred to by its order i in the set N , each link is defined by a couple of nodes i and j , and is denoted as (i, j ) or lij . When the graph is directed, then the order of the two nodes is important: lij stands for a link from i to j , and lij = lj i . For a graph G of size N , the number of edges K is at least 0 and at most N(N −1)/2 (when all the nodes are pairwise connected). The graph is said sparse when the number of edges is of the same order of the number of nodes, while it is said dense when the number of its links is of the order of the square of the number of nodes. The following is a brief account of the main quantities and measures that are commonly used to characterize the properties of a complex network. A central concept is that of reachability of two different nodes of a graph. In fact, two nodes that are not directly connected may nevertheless be reachable from one to the other. In particular, a walk from node i to node j is an alternating sequence of nodes and edges (a sequence of adjacent nodes) that begins with i and ends with j . The length of the walk is defined as the number of edges in the sequence. A trail is a walk in which no edge is repeated. A path is a walk in which no node is visited more than once. The walk of minimal length between two nodes is known as shortest path or geodesic. A graph G = (N , L) can be completely described by giving the adjacency matrix A, a N × N square matrix whose entry aij (i, j = 1, . . . , N ) is equal to 1 when the link lij exists, and zero otherwise. The degree ki of the node i is the number of edges incident with that node, and is obviously defined in terms of the adjacency matrix A as:

ki = (6.1) aij . j ∈N

The most basic topological characterization of a graph G can be obtained in terms of the degree distribution P (k), defined as the probability that a node chosen uniformly at random has degree k or, equivalently, as the fraction of nodes in the graph having degree k. The n-moment of P (k) is defined as:

k n = (6.2) k n P (k). k

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The first moment k is the mean degree of G, while the second moment measures the fluctuations of the connectivity distribution. Graphs are usually said homogeneous (heterogeneous) if the value of the second moment of the degree distribution is small (large) if compared with the value of the first moment of the same distribution. Given the degree distribution, one has the complete statistical picture of uncorrelated networks. There are relevant cases, however, where the corresponding networks are correlated in the sense that the probability that a node of degree k is connected to another node of degree, say k , depends on k. In these latter cases, it is therefore necessary to introduce the conditional probability P (k |k), being defined as the probability that a link from a node of degree k points to a node of degree k . For uncorrelated graphs, in which P (k |k) does not depend on k, we have P (k |k) = k P (k )/k. The distribution of geodesics plays a crucial role, as one can imagine, in all processes involving transport of information across the network. It is therefore useful to represent all the shortest path lengths of a graph G as a matrix D in which the entry dij is the length of the geodesic from node i to node j . The maximum entry of the matrix D is called the diameter of the graph, as it indicates the pair of nodes of maximal distance along the corresponding shortest path, and, as so, it gives a measure of the maximal extent of a graph. A measure of the statistically typical separation between any two nodes in the graph is given by the average shortest path length, also known as characteristic path length, defined as the mean of geodesic lengths over all couples of nodes [242]:

1 dij . L= (6.3) N (N − 1) i,j ∈N , i =j

It is extremely relevant in a network the way the characteristic path length scales with the number of nodes. In particular, a graph is said to display the small world property if L scales with the logarithm of N , indicating that, as the network size increases, this does not affect substantially the mean distance between any pair of nodes of the graph. The communication of two nonadjacent nodes, say j and k, depends on the nodes belonging to the paths connecting j and k. Consequently, a measure of the relevance of a given node can be obtained by counting the number of geodesics going through it, and defining the so-called node betweenness. Together with the degree and the closeness of a node (defined as the inverse of the average distance from all other nodes), the betweenness is one of the standard measures of node centrality, originally introduced to quantify the importance of an individual in a social network [243].


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More precisely, the betweenness bi of a node i, sometimes referred to also as load, is defined as [244,245]:

nj k (i) , bi = (6.4) nj k j,k∈N , j =k

where nj k is the number of shortest paths connecting i and j , while nj k (i) is the number The concept of betweenness can be extended also to the edges. The edge betweenness is defined as the number of shortest paths between pairs of nodes that run through that edge [246]. This latter quantity will be used extensively later on when we will discuss suitable weighting strategies able to enhance the synchronization features of networking systems. A further important quantity is the graph clustering coefficient C, which is defined as follows. A quantity ci (the local clustering coefficient of node i) is first introduced, expressing how likely aj m = 1 for two neighbors j and m of node i. The value of ci is actually obtained by counting the number of edges (denoted by ei ) in the subgraph of neighbors of i. The local clustering coefficient is defined as the ratio between ei and ki (ki − 1)/2, the maximum possible number of edges that one can encounter in such subgraph [242]: 2ei j,m aij aj m ami ci = (6.5) = . ki (ki − 1) ki (ki − 1) The clustering coefficient of the graph is then given by the average of ci over all the nodes in G: 1

ci . C = c = (6.6) N i∈N

By definition, 0 ci 1, and 0 C 1. Given a graph G(N , L), a community (or cluster, or module) is a subgraph G (N , L ), whose nodes are tightly connected, i.e., cohesive. Since the structural cohesion of the nodes of G can be quantified in several different ways, there are different formal definitions of community structures. Communities are seen as groups of nodes within which connections are dense, and between which connections are sparser [247,248]. A possible definition is the following: G is a community if the sum of all degrees within G is larger than the sum of all degrees toward the rest of the graph [249]. At the end of the present chapter we will see that the synchronization–desynchronization processes in a graph are intimately related to the hierarchy of modules and communities forming the network. A series of important information on the dynamical properties of G can be extracted from the spectrum of the Laplacian matrix.


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The combinatorial Laplacian matrix Λ is defined as Λ = D − A, where D is the diagonal matrix with elements

Dii = aij = ki . j

Λ is, by construction a zero row sum matrix. Furthermore, as far as the original graph is undirected, all eigenvalues of Λ are real and nonnegative, and Λ has a full set of N real and orthogonal eigenvectors. Since all rows of Λ sum to zero, Λ always admits the lowest eigenvalue λ1 = 0, with corresponding eigenvector v1 = (1, 1, . . . , 1). 6.1.2. Weighted graphs The above definitions and measures characterize the main properties of unweighted networks, i.e., networks that have a binary nature, where the edges between nodes are either present or not. As we shall see, enhancement of synchronization usually requires the networks to be weighted, i.e., each link is associated with a real number which measures the relative strength of the connection. This, indeed, reflects what happens in many real world networks, where, along with a complex topological structure, a large heterogeneity in the capacity and the intensity of the connections is present. It is important, therefore, to reformulate all the above mentioned quantities for the weighted network case. A weighted graph GW = (N , L, W) consists of a set N = {n1 , n2 , . . . , nN } of nodes, a set L = {l1 , l2 , . . . , lK } of links, and a set of weights W = {w1 , w2 , . . . , wK } that are real numbers attached to the links. In matrix representation, GW is usually described by the so-called weights matrix W, a N × N matrix whose entry wij is the weight of the link connecting node i to node j , and wij = 0 if the nodes i and j are not connected. As for unweighted graphs, an adjacency matrix A can be constructed, represents the underlying wiring of connection. This is done by fixing aij = 1 if wij = 0, and aij = 0 if wij = 0. The natural generalization of the degree ki of a node i is the node strength (or node weight, or node weighted connectivity) si , defined as [250–252]:

si = (6.7) wij . j ∈N

The strengths integrate the information on the number (degree) and the weights of links incident in a node. When the weights are independent on the topology,


191

the strength of the vertices of degree k is s(k) wk where w is the average weight. For a given node i, with connectivity ki and strength si , different situations can arise. All weights wij can be of the same order si /ki or, in contrast, only one or a few weights can dominate over all the others. This fact naturally leads to introduce the concept of node disparity. The disparity in the weights of a node i can been evaluated [253,254] by the quantity Yi defined as [255,256]:

wij 2 Yi = (6.8) , si j ∈Ni

where Ni is the set nodes that are linked in the graph with the node i. If all the edges have comparable weights, then Y (k) (i.e., the disparity averaged over all nodes with degree k) scales as 1/k. If, however, the weight of a single link dominates equation (6.8), then Y (k) 1; in other words Y (k) is independent of k [254]. The strength distribution R(s) measures the probability that a vertex has strength s, and, altogether with the degree distribution P (k), provides useful information on a weighted network. In particular, since the node strength is related to the node degree, it is expected to observe heavy-tailed R(s) distributions in weighted networks with slow decaying P (k). In a generic weighted network, the length of the edge can be introduced as some function of the weight. For instance it could be taken ij

= 1/wij .

This makes that, given a pair of nodes, the shortest path (in the sense of the path with the minimum number of edges) connecting the nodes is not necessarily the one with minimal distance. It is then necessary to introduce a definition of weighted shortest path length dij as the smallest sum of the edge lengths throughout all the possible paths in the graph from i to j . As for the weighted clustering coefficient, in Ref. [250], Barrat et al. have defined this new quantity for a given node i as:

(wij + wim ) 1 aij aj m ami . ciW = (6.9) si (ki − 1) 2 j,m

These few definitions and concepts will help us in proceeding with the description of the main synchronization processes that might take place in such a complex (weighted or unweighted) graphs. Of course, they do not exhaust the number of

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definitions and measures that characterize the topology and statistics of complex networks. If the reader is interested to have a much more detailed description of concepts, ideas and measures for the characterization of a complex networks, it is worth mentioning that a variety of monographic articles [240,241,257–259] and books [260–263] have already appeared in the literature, and they constitute a very useful source to gather a complete overview on the subject.

6.2. Coupling schemes in lattices It is clear that, with the help of the above definitions and measures, one can approach the problem of synchronization in arbitrarily coupled systems. In this section we start with a descriptive account of the case of dynamical units that are arranged on a generic D-dimensional lattice. Here, one has to distinguish between three types of coupling schemes that can be considered: global coupling where each unit interacts with all the others, local coupling where an element interacts with its neighbors (defined by a given metric), and nonlocal or intermediate couplings. In an ensemble of globally coupled oscillators, the structure of connections is not dependent on the spatial distance between the oscillators. Actually, in this case, one cannot even speak of having dynamical units arranged on a spatial lattice, as each individual unit is influenced by the dynamics of all the other units through an interaction of a mean-field type. In ensembles of both limit-cycle and chaotic oscillators with slightly different oscillation modes, many studies have individuated a phase transition associated to a collective and coherent behavior that may be observed for strong enough coupling strength [66,264–266]. In an opposite approach, several studies considered oscillators embedded in D-dimensional lattices, where each unit interacts only with its nearest neighbors. In this latter case, although different collective regimes can be observed (global or partial synchronization, anti-phase synchronization, phase clustering, coherence resonance) in large ensembles of coupled chaotic or periodic elements, cooperative dynamics are in general dependent on the size and dimension of the lattices, as well as on the distribution of the eigenfrequencies of each oscillator [267–274]. Inspired from some biological systems, where the cell-to-cell interaction is mediated by rapidly diffusive chemical transmitters, some schemes have been proposed to take into account the natural decay of the information content with the distance. In both theoretical and experimental studies, the dependence of the spatial correlation on the range of nonlocal coupling was found to decay with a power law [275–278].


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Other long-range interaction schemes include the ones, in which the interaction strength decays with the lattice distance following a power law r −α where r is the distance between the oscillators and α 0 defines the range of the coupling: the limit α → 0 corresponds to the nearest neighbor coupling, whereas an uniform coupling is obtained for α = 0. In large networks of limit-cycle or chaotic oscillators, power law coupling schemes were found to yield collective behaviors which also depend on the range of coupling (given by the parameter α) and the distribution of the modes (natural frequencies) of each unit [279–282]. In the following, however, the main focus will be that of describing situations where the coupling between dynamical units is ruled by a complex topological wiring of connections, rather than arranging the units on top of some spatial lattice.

6.3. The master stability function We have already discussed in Section 2.6 the issue of assessing the conditions for the stability of the synchronous behavior for a generic coupling topology and configuration. We here remind the main ideas related to this subject, insofar as they will be useful in the study of specific network configurations. Let us consider a generic network of N coupled dynamical units, each one of them giving rise to the evolution of a m-dimensional vector field xi ruled by a local set of ordinary differential equations x˙ i = Fi (xi ). The equation of motion reads: x˙ i = Fi (xi ) − σ

N

Gij H[xj ],

i = 1, . . . , N.

(6.10)

j =1

Here x˙ i = Fi (xi ) governs the local dynamics of the ith node xi ∈ Rd , the output function H[x] is a vectorial function, σ is the coupling strength, and G is a coupling matrix, accounting for the topology of the network wiring. In order to proceed with the analytic treatment, we will make the following explicit assumptions: (i) The network is made of identical (Fi (xi ) = F(xi ), ∀i) and autonomous systems. (ii) The coupling matrix Gij is a real zero-row sum matrix, i.e., the diagonal elements are given by Gii = − j =i Gij . For the time being, the matrix G is assumed to be symmetric (and thus diagonalizable), and therefore there exists a set of eigenvalues λi (of associated orthonormal eigenvectors [vi ]), such that Gvi = λi vi and vTj ·vi = δij . Further, because the zero row condition, the


194

spectrum of eigenvalues is entirely semi-positive, i.e., λi 0 ∀i. However, the entire formalism has been recently extended to the case of nondiagonalizable matrices [283], and therefore the main results that will be stated are generically valid for any kind of network’s topology. Taken together, these assumptions ensure that the coupling term in equation (6.10) vanishes, making the synchronized state an invariant manifold. The synchronized state xi (t) = xs (t), ∀i with x˙ s = F(xs ) is a—possibly unstable— solution of equation (6.10), and therefore stability of the synchronous state reduces to take care of the system’s dynamical properties along all directions in phase space that are transverse to the synchronization manifold. A major advancement in the linear stability analysis of equation (6.10) for the synchronized state was the observation that the analysis can be divided into a topological part and a dynamical part [43]. The topological part involves the calculation of the spectrum of the coupling matrix G, while for the dynamical part the Master Stability Function for F and H needs to be obtained. A necessary condition for stability of the synchronization manifold [14,15] is that the set of (N − 1) ∗ m Lyapunov exponents that corresponds to phase space directions transverse to the m-dimensional hyperplane x1 = x2 = · · · = xN = xs be entirely made of negative values. Let δxi (t) = xi (t) − xs (t) = δxi,1 (t), . . . , δxi,m (t) be the deviation of the ith vector state from the synchronization manifold, and consider the m × N column vectors X = (x1 , x2 , . . . , xN )T and δX = (δx1 , . . . , δxN )T . The equation of motion for the perturbation δX can be straightforwardly obtained by expanding equation (6.10) in Taylor series of 1st order around the synchronized state, which gives

˙ = IN ⊗ JF(xs ) − σ G ⊗ JH(xs ) δX, δX (6.11) where ⊗ stands for the direct product between matrices, and J denotes the Jacobian operator. One can notice that the arbitrary state δX can be written as δX =

N

i=1

vi ⊗ ζi (t)

6.4. Key elements for the assessing of synchronizability

195

[ζi (t) = (ζ1,i , . . . , ζm,i )]. By applying vTj to the left side of each term in equation (6.11), one finally obtains a set of N blocks (the variational equations in an eigenmode form) for the coefficients ζi (t) that read ζ˙j = Kj ζj ,

(6.12)

where j = 1, . . . , N and Kj = [JF(xs ) − σ λj JH(xs )] is the evolution kernel. It is important to notice that each block in equation (6.12) corresponds to a set of m conditional Lyapunov exponents (the kernels Kj are calculated on the synchronization manifold) along the eigenmode corresponding to the specific eigenvalue λj . Because the zero-row condition of the coupling matrix, λ1 ≡ 0 with associated eigenvector v1 = √±1 {1, 1, . . . , 1}T entirely defines the synchronization maniN fold. All the other eigenvalues λi (i = 2, . . . , N ) of associated eigenvectors vi span all the other directions of the (m × N )-dimensional phase space transverse to the synchronization manifold. As the eigenmode associated to the eigenvalue λ1 = 0 lies entirely within the synchronization manifold, the corresponding m conditional Lyapunov exponents equal those of the single uncoupled system x˙ = F(x). For an arbitrary diagonalizable coupling matrix, the eigenvalues spectrum is either real or made of pairs of complex conjugates, and the terms σ λi may take, in general, complex values [49]. We can therefore consider the following d-dimensional parametric equation ζ˙ = JF(xs )ζ − (α + iβ)H[ζ ],

(6.13)

that, once coupled with the d-dimensional local nonlinear evolution of the networked system (˙xs = F(xs )), gives a set of d Lyapunov exponents for any choice of α and β. The surface (Λ(α, β)) in the parameter plane tracing the maximum of such exponents is called Master Stability Function [43], and fully defines the stability properties of the synchronization manifold. If all the eigenmodes (associated with the different λi , i 2) are stable (Λ < 0) the synchronous state is then stable.

6.4. Key elements for the assessing of synchronizability The main strategies for assessing the synchronizability of a given coupling configuration have been described in all details in Section 2.7. In the following we will see how these strategies can be implemented for specific network topologies in order to individuate optimal coupling configurations that can lead to enhancing the arousal of synchronization.


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A series of early studies on this subject have approached the issue of relating the stability of the synchronization manifold to the specific topological properties such as the heterogeneity of the degree distribution, the average path length and the betweenness centrality [284–286]. In particular, in Refs. [284,287], bounds for the eigenratio λλN2 of the Laplacian matrix of networks with different topologies have been derived based on graph theory concepts. The obtained expressions depend solely on graph theoretic quantities such as the degree distribution, the shortest path length, diameter of the coupling graph, or the number of shortest path lengths passing through the edges (edge’s load). These works initiated a flurry of research trying to relate the stability of the synchronization manifold to purely topological properties. For instance, in the case of more general weighted networks, very precise bounds for the eigenratio have been recently provided in Refs. [288–290]. Such quantities depend solely on the heterogeneity of the so-called nodes intensities, defined as the input weight sums for each node. Under the extra assumptions of an uncorrelated distribution of nodes degrees and nonnormalized couplings, the synchronizability of an arbitrary network (with a sufficiently random wiring structure) was found to be increased by decreasing the heterogeneity of the nodes intensities. A basic assumption characterizing most of the early works on synchronization in complex networks is that the local units are symmetrically coupled with uniform undirected coupling strengths (unweighted links). A later interest has born on showing how asymmetric weighted coupling configurations may enhance synchronization of complex networks. Recent studies have revealed the strong influence of weighted and asymmetric coupling configurations on the emergence of coherent global behavior. 6.4.1. Coupling matrices with a real spectra In an initial work [288], the authors proposed a weighting approach based on degree of the node ki . Namely, the connections between nodes are weighted by the node degrees such that the coupling term of equation (6.10) becomes N σ

β

ki

Lij H[xj ],

(6.14)

j =1

where Lij is the usual Laplacian matrix with diagonal entries Lii = ki and offdiagonal entries Lij = −1 if nodes i and j are connected by a link, and Lij = 0 otherwise. Weights of the connections are tuned by the parameter β (with β = 0 recovering the unweighted and undirected configuration). Although the resulting coupling matrix G is asymmetric for all β = 0, it can −β −β be written as a product G = DL, where D = diag{k1 , . . . , kN }. Using matrix


197

identities [49], one can show that the eigenvalue spectrum of G coincides with the spectrum of the matrix W = D 1/2 LD 1/2 , and therefore is real with nonnegative values. A large class of networks weighted with this approach have an optimal condition for synchronization (the minimum of the eigenratio as a function of β) at β = 1 [288]. It is worth to notice that the condition β = 1 corresponds to a situation where the input strength of the coupling term is equal for all the nodes (the diagonal elements of the coupling matrix are normalized to 1), while, for β = 1 such elements are not normalized and are, in general, size-dependent. The relevant results of Ref. [288] is the tremendous improvement on the propensity of synchronization, obtained with a weighting procedure that retains information on the local features of the network (the node degree). Further analysis on directed and weighted networks with criteria based on the overall cost of the coupling has been carried out by the same authors in Refs. [289,290]. We can notice that the weighting procedure proposed in equation (6.14) uses only locally available information of the network (the node’s degree). The problem of how a weighting procedure can enhance the propensity for synchronization has been studied also in the framework of properly normalized coupling schemes making use of the information contained in the overall topology [292], showing that this can give rise to further improvements in the synchronization properties. It turned out to be advantageous to weight a connection with its load. The load ij of the link connecting nodes i and j quantifies the number of shortest paths that are making use of that link. Precisely, for each pair of nodes i , j in the network, we count the number n(i , j ) of shortest paths connecting them. For each one of such shortest paths, we then add 1/n to the load of each link forming it. It is important to notice that the load distribution retains full information on the network structure of pathways at a global level, since the value of each load ij can be strongly influenced also by pairs of nodes that may be very far away from either nodes i and j . Furthermore, it is essential to highlight that this weighting procedure is totally different from relying on the information on the node degrees, insofar as nodes with low degrees in the networks may be connected through links with very high loads, whereas nodes with high degrees may have links with very poor loads. With this approach, the coupling term of equation (6.10) reads

σ α (6.15) ij H[xi − xj ], α j ∈Ni

ij j ∈Ni

where α is a real tunable parameter, and Ni is the set of neighbors to the ith node. The coupling matrix has now several properties: first, the diagonal elements of G are always normalized to 1. Second, as for the weighting approach based on the node degrees, G is asymmetric for all α but it can be written as a product


198

G = BL, where L is a zero row-sum matrix with off-diagonal entries Lij = − and

α ij ,

1 B = diag j

α 1j

,...,

1 j

α Nj

.

The spectrum of G is therefore the same as that obtained from the matrix W = B 1/2 LB 1/2 , i.e., real and with nonnegative values. Because G has a zero row-sum and the network is connected, the smallest eigenvalue λ1 is zero while 0 < λk 2 for k 2 [49]. Another important point to be stressed concerns the various limits that the coupling term can assume when varying α. The best condition for synchronization of Ref. [288] is recovered for α = 0. The limit α = +∞ (α = −∞) induces a unidirected tree structure in the wiring (a network with N nodes and N directed links), such that only the link with maximum (minimum) load is selected as the incoming link for each node. The resulting network will be either connected or disconnected. In the connected (disconnected) case, the ratio λN /λ2 will be equal to 2 (+∞), thus yielding the optimal (the worst) condition for synchronization. 6.4.2. Numerical simulations With the above weighting approaches, one can evaluate the propensity for synchronization of the following classes of complex networks: (i) Scale Free networks. The used class of scale-free networks is obtained by a generalization of the preferential attachment growing procedure introduced in Ref. [293]. Namely, starting from m + 1 all to all connected nodes, at each time step a new node is added with m links. These m links point to old nodes with probability ki + B , j (kj + B)

pi =

where ki is the degree of the node i, and B is a tunable real parameter, representing the initial attractiveness of each node. The advantage of this procedure is that it allows a selection of the γ exponent of the power law scaling in the degree distribution (p(k) ∼ k −γ (B,m) ) B in the thermodynamic (N → ∞) limit. While the with γ (B, m) = 3 + m average degree is by construction k = 2m (thus independent on B), the heterogeneity of the degree distribution can be strongly modified by B. Notice that the case B = 0 recovers the preferential attachment rule originally introduced in Ref. [293].


199

(ii) Small World networks. Small world networks were obtained via the rewiring procedure proposed in Ref. [242]. We start from a ring lattice with N nodes connected with their k nearest neighbors, and each edge is rewired at random with probability p. Self and duplicate connections are forbidden. This procedure provides a class of networks with a highly homogeneous degree distribution. The limit of regularity is obtained for p = 0, whereas the small world regime lies in the intermediate region 0 < p < 1. Fully random networks are obtained with p = 1, for which all nearest neighbor connections are randomly rewired to other nodes in the network. In all cases, the value of k was chosen such that the average degree is k = 2m. Figure 6.1(a) shows the logarithm of λN /λ2 in the parameter space (α, B) for Scale Free networks. The first crucial observation is that the surface of λN /λ2 has a pronounced minimum at 0 < α˜ 1 for all values of B above a given Bc > 0. For the case Bc = 0, the eigenratio decreases as the parameter α increases. Figure 6.1(b) reports the behavior of log(λN /λ2 ) as a function of (α, p), for the Small World regime. Here, the ability for synchronization increases with the randomness of connections. For all α random configurations (p = 1) provide topologies with higher synchronizability than small world or regular networks. However, the eigenratio always displays a minimum at α˜ 1 in all the range of values 0 p 1. For a large class of networks (regular lattices, Small World networks, random wirings and Scale Free networks with Bc > 0), large values of α have a negative effect in the ability of the network to synchronize. The reason for this is that increasing α above α˜ makes that a structure similar to a directed tree dominates the network connectivity. From one side this tendency generically increases the synchronizability of the graph, from the other side it also increases the chance for the network to be disconnected. One has to expect that a second critical value αc > α˜ exists such that the disconnection mechanism dominates over the tree structure induction, determining a global decrease of graph synchronizability. All this discussion has focused on m = 2, which is the minimal value of m for which the resulting graph contains loops and cycles. If one increase m, log(λN /λ2 ) is reduced for all the kind of networks and also α˜ decreases. For the scale free and regular topologies, however, augmenting m also increases the likelihood of disconnecting the network in the limit α → ∞. As a particular case, m = 1 always ensures that scale free networks remains connected at α → ∞, regardless of B. Finally, one should notice that the condition α = 0 recovers the optimal condition (the condition β = 1 in Ref. [288]) when only the local information on node degrees is used in the weighting process. Because the optimal condition based on an edge betweenness weighting procedure is for α˜ 1 for a large class of networks, this indicates that a weighting

200


Figure 6.1. Eigenratio λN /λ2 (in logarithmic scale) vs. the dimensionless parameter space (α, B) for scale free networks (a), and (α, p) for small world networks (b). In all cases m = 2, and the reported values refer to an average over 10 realizations of networks with N = 1000 nodes. Notice that, in all cases, the eigenratio displays a minimum at α˜ 1.

procedure based on the link loads always enhances the synchronizability of the network. This is illustrated in Figure 6.2, where it is shown a plot of the surface Γ delimiting the area in parameter space where the values of λN /λ2 are smaller than those obtained at α = 0.


201

Figure 6.2. Surfaces Γ obtained as the α = 0 crossing of the surfaces λN /λ2 (in logarithmic scale). These surfaces are sketched in the dimensionless parameter space (α, B) for scale free networks (a), and (α, p) for small world networks (b).

For both the Scale Free and Small World networks, one can easily see that for all topological configurations (B or p for the SF or SW networks, respectively) there exists a large parameter interval α ∈ [0, αc ] with Γ (α) < Γ (0). In this region the weighting procedure based on the global load information provides a


202

better synchronization propensities than a weighting process based on the local node degrees. 6.4.3. Coupling matrices with a complex spectra Both the approaches developed in Refs. [288,292] were dealing with situations wherein the coupling matrix G had a real spectrum of eigenvalues. Motivated by what happens in several relevant cases of natural networks [294, 295] (where interactions between individuals are not symmetric), Ref. [296] analyzed networks of asymmetrically coupled dynamical units, where asymmetry was explicitly related to an age order among different nodes. In practice, the idea of Ref. [296] was that the direction of an edge can be determined by an age ordering between the connected nodes. For instance, in growing networks, such age ordering will be naturally related to the appearance order of the node during the growing process. The implementation of this idea leads to consider in equation (6.10) a zero row-sum coupling matrix Gij with off diagonal entries: Gij = −Aij

Θij j ∈Ni

Θij

,

(6.16)

where A is the adjacency matrix, and 1−θ 2 for i > j , while Θij =

Θij =

1+θ 2

for i < j . Furthermore, as usual, Ni denotes the set of ki neighbors of the ith node. The parameter −1 < θ < 1 has here the crucial role of governing the coupling asymmetry in the network. Precisely, θ = 0 yields the optimal synchronization condition of Ref. [288], being

1 = ki , j ∈Ni

while the limit θ → −1 (θ → +1) gives a unidirectional coupling configuration wherein the older (younger) nodes drive the younger (older) ones. Though being asymmetric, for θ = 0 the matrix G has a real spectrum of eigenvalues, and the results are the same as those obtained in Ref. [288] for β = 1. Conversely, for a generic θ = 0, the coupling matrix has a spectrum contained in the complex plane (λ1 = 0; λl = λrl + j λil , l = 2, . . . , N ). Furthermore, by construction, the diagonal elements of G are normalized to 1 in all possible cases.


203

As discussed above, if R is the bounded region in the complex plane where the master stability function provides negative Lyapunov exponents, the stability condition for the synchronous state is that the set {σ λl , l = 2, . . . , N } be entirely contained in R for a given σ , which is best realized when both the ratio λrN λr2 and M ≡ max λil l

are simultaneously made as small as possible. With this in mind, Ref. [296] analyzed the effects of heterogeneity in the node degree distribution, by comparing the propensity for synchronization of the class of scale free networks explained above, with that of a highly homogeneous Erdös– 2m (giving the same Rényi random network, having connection probability P = N−1 average degree k = 2m), and an arbitrary initial age ordering. λr Figure 6.3(a) reports the behavior of the eigenratio λNr vs. θ for scale free 2 networks (solid line) and for the random topology (dashed line). It is possible to notice that while the best synchronizability condition for random networks is θ = 0, scale free networks show a better (worse) propensity for synchronization for θ → −1 (θ → 1). The imaginary part of the spectra M vs. θ illustrated in Figure 6.3(b) indicates that only very small differences exist between the scale free and the random network configurations in the whole range of the asymmetry parameter. An important point illustrated in Figure 6.4(a) is that the monotonically deλr creasing behavior of λNr with θ persists for different degree of heterogeneity 2 (tuned by the parameter B), indicating that synchronization is always enhanced in growing scale free networks when θ becomes smaller. Similarly, results depicted in Figure 6.4(b) highlight that the contribution to network synchronizability of the imaginary part of the spectra does not depend significantly on the specific value of B. Ref. [296] further elaborated these results, suggesting that the effect of enhancing the synchronization in weighted and directed complex networks is the result of the simultaneous presence of two ingredients. The first one is that the weighting must induce a dominant interaction from hub to nonhub nodes. Indeed, for positive (negative) θ values, the dominant coupling direction is from younger (older) to older (younger) nodes. Now, in growing scale free networks the minimal degree of a node is by construction m and older nodes are more likely to display larger degrees than younger ones, so that a negative θ here induces a dominant coupling direction from hubs to nonhub nodes.

204


Figure 6.3. Panel (a): Eigenratio λN /λ2 (in logarithmic scale); panel (b): M vs. the tuning parameter θ for scale free networks (solid line) and random networks (dashed line). In all cases the reported values refer to an average over 10 realizations of networks with N = 1000 nodes.

A second fundamental ingredient is that the network contains a structure of connected hubs influencing the other nodes. This is in general accounted for by a proper normalization in the off diagonal elements of coupling matrix G, assuring that hubs receive an input from a connected node scaling with the inverse of their degree, and therefore the structure of hubs is connected always with the rest of the network in a way that is independent on the network size.


Figure 6.4.

205

Panel (a): Eigenratio λN /λ2 (in logarithmic scale); panel (b): M vs. the dimensionless parameter space (θ, B) for scale free networks.

As an example, for age ordered growing scale free networks, the nonzero off diagonal elements of the coupling matrix are given by: Cij =

1∓θ , ki − θ [ki − 2 min (i − 1, m)]

(6.17)

where ∓ stands for i > j and i < j , respectively. The recent works of [288,289,292,296,290] have shown that weighted networks is the most promising framework for the study of how the architectural properties of the connection wiring influence the emergence of a collective (syn-


206

chronized) behavior, in the case in which the connection wiring is constant in time.

6.5. Networks with degree–degree correlation Ref. [297] made a step ahead, and studied synchronizability of graphs in the presence of degree–degree correlations. This study was motivated by the observation that almost ubiquitously real networks feature particular forms of correlation or mixing among the network vertices. In such a latter case, the topology of the wiring cannot be completely described by the degree distribution, and it has to be expected that these correlation effects contribute in a nonnegligible way to the dynamical processes taking place on top of such networks. One form of mixing is the correlation among pairs of linked nodes according to some properties at the nodes. A very simple case is the degree–degree correlation, in which vertices choose their neighbors according to their respective degrees. This property can be conveniently measured by means of a single normalized index, the Pearson statistic r defined as follows: 1 kk (ekk − qk qk ), r= 2 (6.18) σq k,k

where qk is the probability that a randomly chosen edge is connected to a node having degree k; σq is the standard deviation of the distribution qk and ekk represents the probability that two vertices at the endpoints of a generic edge have degrees k and k respectively. Positive values of r indicate assortative mixing (i.e., the tendency of nodes to form connections with their connectivity peers), while negative values characterize disassortative networks (i.e., the tendency of nodes with high degree to be linked to nodes with low degrees). Ref. [297] seeks to characterize how the combination of variable degree correlation, and variable asymmetry over the network weights may indeed affect the network synchronizability. In order to construct networks, characterized by a given degree distribution, Ref. [297] used the so-called configuration model. Specifically this consists by constructing a network of N vertices by means of the following procedure: • One starts by assigning a desired form of the degree distribution P (k). • One assigns to each of N vertices a target degree k > kmin drawn from the distribution P (k), and attributes to that vertex a number k of half edges.

6.5. Networks with degree–degree correlation

207

• One randomly connects pairs of half edges into an edge, until no half edges remain, and all nodes have reached their target degree. This process leads to a final random graph with the desired degree distribution P (k), which, however, does not show any degree-degree correlation property. In order to reproduce a desired level of degree correlation, Ref. [297] performed target exchanges of links between pairs of connected nodes, until the desired value of the observable r is reached. Furthermore, Ref. [297] considered all the nodes of the network as being connected asymmetrically, with the same approach described in the previous section. As a result, Ref. [297] could evaluate the changes of the network synchronizability in the parameter space (r, θ). The main obtained results are shown in Figures 6.5 and 6.6, for scale free networks with degree distribution exponent γ = {2, 3}. The effects on the eigenratio R ≡ λN /λ2 are reported in Figure 6.5. Note that R is an increasing function of θ for all the values of r considered. This depends on the advantage, in terms of the network synchronizability, of having asymmetric interactions directed from high-degree nodes to low-degree ones. Moreover this is particularly evident for low values of θ in the case of assortative networks. Thus the combination realized when θ → −1 and the network topology is assumed to be strongly assortative (high r) represents the optimum configuration for the minimization of R. This is also consistent with the claim that the optimal network configuration can be realized, when both a dominant interaction from high-degree nodes to low-degree ones and a structure of interconnected hubs are present. On the other hand, when θ ≈ 0, one observes a completely different picture with disassortative networks being characterized by better synchronizability properties. The same behavior is also confirmed in the case of positive values of θ, where the better performance of disassortative networks is even enhanced. Thus the onset of two separate regimes emerges as varying θ: the first one, in the case where there is a dominant interaction of the high-degree nodes on the lowdegree ones, with assortative mixing representing a desirable property in terms of the network synchronizability; the second one, with symmetric coupling or directed from low-degree nodes to high-degree ones, where disassortative mixing represent the best choice in order to enhance the network synchronizability. In Figure 6.6 it has been shown the behavior of the maximum imaginary part J of the spectrum as varying θ . As expected, this is characterized by a minimum at θ = 0 (for which the spectrum of the coupling matrix is real) and increases as the asymmetry over the network links increases. Note that for all the values of θ = 0 assortative (positively correlated) networks are characterized by higher values of J when compared to their uncorrelated and disassortative counterparts.

208


(a)

(b) Figure 6.5. Reprinted with permission from Ref. [297]. Eigenratio λN /λ2 as function of q for scale free networks characterized by variable degree correlation: r = [−0.3, −0.15, 0, 0.15, 0.3] (the corresponding symbols are indicated in the figure). (a) γ = 2, N = 1000, (b) γ = 3, N = 1000.

6.5. Networks with degree–degree correlation

209

(a)

(b) Figure 6.6. Reprinted with permission from Ref. [297]. Maximum imaginary part of the spectrum of L as a function of q for scale free networks characterized by variable degree correlation: r = [−0.3, −0.15, 0, 0.15, 0.3]. (a) γ = 2, N = 1000, (b) γ = 3, N = 1000.


210

6.6. Synchronization in networks of phase oscillators After having discussed the general stability properties of synchronization states in complex networks, we pay attention now to summarize the most significant obtained results in the study of synchronization of networking oscillators. The very first network models considered ensembles of limit cycle oscillators, each one of them was ruled by the Kuramoto dynamics, since this latter situation allows for both analytical and fast numerical simulations approaches. Such works studied a generic graph where each node i (i = 1, . . . , N ) is a planar rotor characterized by an angular phase, θi , and a natural or intrinsic frequency ωi . In such a model, two oscillators interact if they are connected by an edge of the underlying graph. The individual dynamics of the ith node is described by:

sin(θj − θi ) θ˙i = ωi + σ (6.19) j ∈Ni

where Ni is the set of neighbors of i, and σ is the coupling strength, identical for all edges. The set of natural frequencies and the initial values of θi are in general randomly chosen from a given distribution [66]. The original Kuramoto model corresponds to the simplest case of globally coupled (complete graph) equally weighted oscillators, where the coupling strength is taken to be σ = ε/N in order to warrant the smoothness of the model behavior also in the thermodynamic limit N → ∞ [66]. In this case, the onset of synchronization occurs at a critical value of the coupling strength εc = 2/πg(ω0 ), where g(ω) is the distribution from which the natural frequencies are drawn, and ω0 represents the mean frequency of the ensemble. The transition to synchronization is a second-order phase transition characterized by the order parameter: N 1

iθj (t) e r(t) = (6.20) . N j =1

When the two limits N → ∞ and t → ∞ are considered (and for ε εc ), the order parameter r behaves as r ∼ (ε − εc )β , with β = 1/2. The mechanism through which synchronization emerges in the system is as follows. As the coupling takes very small values, the strength of the interactions is not enough to break the incoherence produced by the individual dynamics of each oscillator.

6.6. Synchronization in networks of phase oscillators

211

As soon as a critical value σc is overcome, some elements lock their relative phases and a cluster of synchronized nodes comes up. When the coupling is further increased, the population of oscillators is split into a partially synchronized state made up of oscillators locked in phase (that adds to r), and a group of nodes whose natural frequencies are too spread as to be part of the coherent pack. Eventually, after further increasing in σ , more and more nodes get entrained around the mean phase, and the system settles in a completely synchronous state (where r ≈ 1). With the growing interest in complex networks, several groups have moved the attention to investigate the synchronization phenomena of the Kuramoto model in complex wirings. Refs. [298,299] have studied numerically the conditions for the onset of synchronization in random scale-free networks. Specifically, they studied the Kuramoto model on top of scale free networks and on top of small structures (motifs) that were relevant in different biological and social networks, with the aim of inspecting the critical point associated to the onset of synchronization, i.e., when small groups of synchronized oscillators first appear in the system. The onset of synchronization (quantified by the order parameter given by equation (6.20)) for scale free networks was reported to occur at a small, though nonzero, value of the coupling strength, with a critical exponent around 0.5. In contrast with the all-to-all coupling configuration, for the complex topologies the critical point does not depend on the size of the system N . Moreover, as it was discovered later on, the choice of the order parameter seems to be a crucial point when analyzing the conditions for the existence of the transition threshold. Soon afterwards, several other authors [300,301] have investigated the same problem from a theoretical perspective, as well as with numerical simulations. The results do not fully clarify whether or not the critical point exists. The main difficulty comes from the fact that there is no a unique consensus about the set of differential equations describing the system dynamics, and what the order parameter should be. Ref. [300] reported the lack of a critical point in power law graphs when 2 < γ < 3, which is recovered when the second moment of the distribution converges (that is for γ > 3). No estimates can be done for γ = 3 since there the relevant parameters of the system diverge. The same qualitative behavior was reported in Ref. [301], where different analytical approaches were introduced. The mean field theory [300,301] predicts that the critical point is determined by the all-to-all Kuramoto value, σ0 , rescaled by the ratio between the first two moments of the degree distribution: k . k 2 It is worth noticing that Refs. [298,299] have used the classical order parameter of equation (6.20), while the analysis of Refs. [300,301] made use of a rescaled σmf = σ0


212

parameter, given by: N 1

r(t) = kl eiθj (t) . l kl

(6.21)

j =1

In principle, there are reasons for the use of either one of the two parameters. Equation (6.20) does not assign weights to the different connectivity classes, while equation (6.21) incorporates in the definition of the order parameter the connectivity of each node, which at first glance seems to be reasonable. Finally, we briefly refers to clustering and modular synchronization of oscillator networks. This issue, indeed will be extensively described later one in this same chapter, when we will try to show how synchronization features can help to reveal hierarchical topological structures (modules or community structures) hidden in the wiring of connection of a graph. It is indeed clear that, as the route to complete synchronization is made up of groups of synchronized oscillators that grow and coalesce, one would be able to detect structural properties of the underlying network by just a fine tuning in the coupling strength. The effects of clustering have also been studied on different types of modular complex networks, finding that the synchronization transition crucially depends on the type of inter-modular connections. The relaxation time τ for synchronization of hubs was found to be shorter than that of less connected nodes [298]. In particular, for scale free networks τ ∼ k −1 . Hence, the more connected a node is, the more stable it is. Such power-law behavior points to an interesting result, namely, it is easier for an element with high k to get locked in phase with its neighbors than for a node linked to just a few others. Furthermore, the destabilization of a hub does not destroy the synchrony of the group it belongs to. On the contrary, the group formed by the hub’s neighbors recruits it again. All these results indicate that it may be possible to use synchronization phenomena to unravel highly clustered structures embedded within the wiring of a complex networks.

6.7. Synchronization in dynamical networks So far, all the discussion was limited to the case of emerging synchronization in static networks, e.g., networks whose wiring of connections is fixed, or grown, once forever. Recently, a series of important studies have considered the very opposite limit of blinking networks [304,305], where the wiring of connections is rapidly (i.e., with a characteristic time scale much shorter than that of the networked system’s dynamics) switching among different configurations.

6.7. Synchronization in dynamical networks

213

In this latter condition, a relevant result shown in Refs. [304,305] is that synchronous motion can be established for sufficiently rapid switching times even in the case in which each visited wiring configuration would prevent synchronization under static conditions. None of these two limits, however, seems a convenient description of many relevant cases. For instance, properly modelling processes such as mutation in biological systems [306], synaptic plasticity in neuronal networks [307], or adaptation in social or financial market dynamics [308] would require accounting for time varying networks whose evolution takes place over characteristic time scales that are commensurate with respect to the intrinsic one of the nodes’ dynamics. In this section, we follow what was reported in Ref. [309] that went beyond the limit of static wirings, and assessed the conditions for the appearance of synchronized states in dynamical networks, without making any explicit hypothesis on the time scale responsible for the variation of the coupling wiring. Ref. [309] considered a network of N coupled identical systems, whose evolution is described by x˙ i = f(xi ) − σ

N

Gij (t)h[xj ],

i = 1, . . . , N.

(6.22)

j =1

Here again x ∈ Rm is the m-dimensional vector describing the state of the ith node, f(x) : Rm → Rm governs the local dynamics of the nodes, h[x] : Rm → Rm is a vectorial output function, σ is the coupling strength, dots stay for temporal derivatives, and Gij (t) ∈ R are the time varying elements of a zero row-sum ( j Gij (t) = 0 ∀i and ∀t) N × N symmetric connectivity matrix G(t) with strictly positive diagonal terms (Gii (t) > 0 ∀i and ∀t) and negative off diagonal terms (Gij (t) 0 ∀i = j and ∀t), specifying the evolution in strength and topology of the underlying connection wiring. Being G(t) symmetric at all times, it admits always a set λi (t) [vi (t)] of real eigenvalues (of associated orthonormal eigenvectors), such that G(t)vi (t) = λi (t)vi (t) and vTj · vi = δij . The zero-row condition ensures that: (i) the spectrum is entirely semi-positive, i.e., λi (t) 0 ∀i and ∀t; (ii) λ1 (t) ≡ 0 with associated eigenvector v1 (t) = √1 {1, 1, . . . , 1}T that entirely defines a synchronization manifold (xi (t) = xs (t), N ∀i), whose stability was the object of the study reported in Ref. [309]; and (iii) all the other eigenvalues λi (t) (i = 2, . . . , N , λi (t) > 0 for connected graphs) have associated eigenvectors vi (t) spanning the transverse manifold of xs (t) in the (m × N )-dimensional phase space of equation (6.22).


214

Let

δxi (t) = xi (t) − xs (t) = δxi,1 (t), . . . , δxi,m (t)

be the deviation of the ith vector state from the synchronization manifold, and consider the N × m column vectors X = (x1 , x2 , . . . , xN )T and δX = (δx1 , . . . , δxN )T . Then, in linear order of δX, one has

˙ = IN ⊗ Jf(xs ) − σ G(t) ⊗ Jh(xs ) δX, δX

(6.23)

where ⊗ stands for the direct product, and J denotes the Jacobian operator. By further considering that the arbitrary state δX can be written as δX =

N

vi (t) ⊗ ηi (t)

i=1

[ηi (t) = (η1,i , . . . , ηm,i )], and applying vTj to the left side of each term in equation (6.23), one finally obtains

dηj dvi (t) vTj (t) · = Kj ηj − ηi , dt dt N

(6.24)

i=1

where j = 1, . . . , N and

Kj = Jf(xs ) − σ λj (t)Jh(xs ) . The key point is to notice that equations (6.24) transform into a set of N variational equations of the form dηj = Kj ηj , dt as soon as N

i=1

vTj (t) ·

dvi (t) ηi = 0, dt

i.e., when all eigenvectors are constant in time. This can be realized in two different ways. Namely: either the coupling matrix G(t) is constant, or when, starting from an initial wiring condition G0 = G(t = 0), the coupling matrix G(t) commutes at any time with G0 : G0 G(t) = G(t)G0 ,

∀t.


215

While the former case recovers the usual Master Stability Function approach, that was described in Chapter 2, Ref. [309] focussed on the latter situation, and proposed two different techniques to explicitly construct a commutative evolution. The first consists in noticing that the initial symmetric coupling matrix can be written as G0 = V ΛV T , where V = {v1 , . . . , vN } is an orthogonal matrix whose columns are the eigenvectors of G0 , and Λ0 = diag 0, λ2 (0), . . . , λN (0) is the diagonal matrix formed by the eigenvalues of G0 . From the other side, at any time t, a zero row sum symmetric commuting matrix G(t) can be constructed as G(t) = V Λ(t)V T , where Λ(t) = Λ0 + diag 0, δλ2 (t), . . . , δλN (t) = Λ0 + Λ(t), giving G(t) = G0 + V Λ(t)V T , with the additional constraint N

2 δλk (t)vik −Gii (0)

k=2

to maintain strictly positive all diagonal terms. For instance, when δλk = δlk δλl (l = 1), the elements of G(t) are Gij (t) = Gij (0) + δλl vil vj l , so the constraint on the diagonal term reduces to Gii (t) = Gii (0) + δλl (vil )2 0 (which is always satisfied for δλl > 0).

(6.25)


216

This technique allows one to select δλk = δlk δλl in order to fix a desired strength Gi˜j˜ (t) = −d (d 0) for the element i˜ and j˜ of G(t): δλl = −

d + Gi˜j˜ (0) vil˜ vj˜l

.

Furthermore, since the selection of δλl implies a perturbation to all the other elements vil vj l Gij (t) = Gij (0) − d + Gi˜j˜ (0) , vil˜ vj˜l one has to select the proper l by minimizing the quantity (l) =

˜ j =j˜ i =

i, i,j

vil vj l v v ˜ j˜l il

as a function of l. In this way one is warranted to produce a commutative graph where the desired link has the chosen strength, and all the other links have the smallest perturbation as possible from the original pristine network configuration. A second technique proposed in Ref. [309] consists in starting from a given initial graph G0 , and producing a large set of different realizations of the same graph. This allows one to calculate the probability distribution p(λ0 ) of the nonnull eigenvalues of the set. Then one constructs Λ(t) in equation (6.25) by randomly drawing a set of N −1 eigenvalues (λ1 (t) must be always zero) either within the same distribution or using a uniform distribution between λ2 (0) and λN (0). The former strategy can be realized by, e.g., using the spectrum of a different realization of G0 (henceforth called eigenvalues surrogate method), or by randomly drawing the eigenvalues of G(t) from the distribution p(λ0 ) in an ordered (0 < λ2 (t), . . . , λN (t)) or unordered way. The latter strategy can be realized by randomly picking the eigenvalues (in an ordered or unordered way) from a uniform distribution. Ref. [309] studied study how such procedures modify the main topological structures of the underlying network. In general, the resulting G(t) is a dense matrix that can be associated to a symmetric weighted network, whose weight matrix W (t) has elements Wii (t) = 0, and Wij (t) = |Gij (t)| for i = j .


217

For weighted networks, the fundamental measures that characterizes the topology are the average value of the strength distribution, the clustering coefficient, and the average shortest path. In particular, the distance between two adjacent nodes i and j is given by lij = 1 wij , and the distance along a path {n(1), n(2), . . . , n(m)} can be expressed as Ln(1)→n(m) =

m−1

ln(k)n(k+1) .

k=1

The shortest path connecting two nonadjacent nodes is then ij

= min Li→j , all path

and the network’s average shortest path length is i =j

=

2 N (N − 1)

ij .

i,j

As for the clustering coefficient, one can define a clustering of node i as

2 Wij Wim Wj m , ci = si (N − 2) j,m where si = j Wij is the strength associated to the node i. The average clustering coefficient C is then given by 1

ci , C = N i

while the strength distribution P (s) characterizes the heterogeneity of the network. Table 6.1 reports the results of the mentioned procedures operated on an initial condition G0 having a power law (scale-free) strength distribution P (s) ∼ s −γ , and taken to be the Laplacian matrix of a scale-free network grown as originally introduced by Barabási and Albert in Ref. [293] with m = 5. The corresponding strength distributions are plotted in Figure 6.7. It is relevant to notice that the eigenvalue surrogate method, as well as the method of choosing the eigenvalues from the initial distribution p(λ0 ) in an ordered way do not change the scale-free behavior of the strength distribution (see the thick and dashed lines in Figure 6.7(a)), and provide values for s = sP (s) ds, C and that only slightly deviate from those of the original scale-free network (see Table 6.1). Therefore, both methods provide a convenient way to construct a commuting evolution of the graph that substantially preserves all main topological features of the initial condition.


218

Table 6.1

Statistical properties of the commuting graphs

Initial condition G0 (scale-free) Eigenvalue surrogate method Choosing from p(λ0 ) (ordered) Choosing from p(λ0 ) (unordered) Uniform distribution (ordered) Uniform distribution (unordered)

s

C

9.94 11.6881 12.7042 142.8746 123.6484 521.7568

5.5773e–4 8.4730e–04 9.1235e–4 0.0543 0.0634 0.6125

5.5144 5.2758 5.2325 3.7144 0.9570 1.1098

The table reports average strength s = sP (s) ds, the average clustering coefficient C, and the average shortest path length (see text for definitions and details on the construction of the commuting graphs). In all cases < . . . > refers to an ensemble average over 100 different realizations of networks of size N = 500.

Figure 6.7. Log-log plots of the strength distributions P (s) (see text for definition) for a series of commuting graphs. In panel (a) it is reported P (s) for the initial condition G0 (solid line), for G(t) constructed by the eigenvalue surrogate method (thick solid line), and for Λ(t) obtained by randomly choosing the eigenvalues from the same distribution p(λ0 ) in an ordered (0 < λ2 (t) λ3 (t) · · · λN (t)) way (dashed line). In panel (b), P (s) is reported for the case in which the eigenvalues are randomly chosen from p(λ0 ) in a unordered way (solid line), or chosen from a uniform distribution in an ordered way (thick solid line), and in a unordered way (dashed line). Other stipulations as in the caption of Table 6.1. Reprinted with permission from Ref. [309]. © 2006 The American Physical Society


219

At variance, the methods of choosing the eigenvalues from a uniform distribution (either in an ordered or in a unordered) way (thick solid and dashed lines in Figure 6.7(b)) completely destroy the original scale-free strength distribution, and provide far different values for each one of the measured quantities. An intermediate situation occurs when choosing the eigenvalue spectrum from the initial p(λ0 ) in a unordered way. Here, one still preserves the scale-free behavior of the tail (see the solid line in Figure 6.7(b)), but the main topological features of the underlying network are substantially changed from their initial values. Ref. [309] finally discussed the relevant consequences of such a commutative evolution on the stability of the synchronization manifold. Because of the commuting properties of G(t), equation (6.24) becomes η˙j = Kj ηj

(j = 2, . . . , N).

(6.26)

Replacing σ λj (t) by ν in the kernel Kj , the problem of stability of the synchronization manifold is tantamount to study the m-dimensional parametric variational equation η˙ = Kν η, with

Kν = Jf(xs ) − νJh(xs ) , allowing for graphing the curve of Λmax (the largest of the m associated conditional Lyapunov exponents) vs. ν. When G(t) is evolving through commuting graphs, the eigenvectors are fixed in time, and from equation (6.26) we have that at time T = k dt, the modulus of each eigenmode ηi (i = 1) obeys k ηi (T ) exp Λmax σ λi (n dt) dt . η (0) i

(6.27)

n=0

From (6.27) it follows immediately that the condition for transverse stability of the synchronization manifold is that ∀i = 1 1 Si = lim T →∞ T

T

Λmax σ λi (t ) dt < 0.

0

It is worth to notice that (6.28) does not necessarily imply that Λmax σ λi (t) < 0

(6.28)


220

at all times. Rather, one can even construct a commutative evolution such that at each time there exists one eigenvalue λi (i = i(t)) for which Λmax σ λi (t) > 0, and yet obtain a transversally stable synchronization manifold. An example is a periodic evolution with period Tp = (N − 1)τ , during which G(t) is given by G(t) =

N−1

Gl χ[(l−1)τ,lτ ) ,

(6.29)

l=1

with χ[(l−1)τ,lτ ) being the characteristic function of the interval [(l − 1)τ, lτ ), and the matrices Gl (starting from a given G1 = V1 Λ1 V1−1 ) are constructed as Gl = V1 Λl V1−1 for l = 2, . . . , N − 1 with Λ1 = diag(0, λ2 , . . . , λN ), Λl = diag(0, λ2,l , . . . , λN,l ), and λj,l = λ[mod(N−1) (j +l−3)]+2 . If, for instance Λmax (σ λ2 ) > 0, then there will always exist a direction in the phase space, along which the synchronization manifold is transversally unstable, but if N−1

Λmax (σ λj ) < 0,

j =2

then condition (6.28) will be satisfied in all directions transverse to the synchronization manifold, making it transversally stable. An example of such an extreme situation is illustrated in Figure 6.8. Here, an initial scale-free network of N = 200 Rössler chaotic oscillators is considered, each one of them obeying equation (6.22) with x ≡ (x, y, z), f(x) = (−y − z, x + 0.165y, 0.2 + z(x − 10)) and h[x] = y, and the evolution of the wiring follows equation (6.29). For σ = 0.03 we have that N−1

j =2

Λmax (σ λj ) −9.158,


221

Figure 6.8. Time evolution of synchronization error (see text for definition) for τ = 0.2 (empty circles), 0.3 (filled circles), 0.4 (empty squares), 0.5 (filled squares), 0.6 (empty diamonds), 0.7 (filled diamonds), 0.9 (empty up triangles), 1.1 (filled up triangles), and 1.5 (empty down triangles). In all cases, points refer to an ensemble average over 5 different random initial conditions for a N = 200 dynamical network of identical chaotic Rössler oscillators (parameters specified in the text). The inset reports Tsync (see text for definition) vs. the switching time τ (open circles refer to different initial conditions, solid line graphs the ensemble average). Reprinted with permission from Ref. [309]. © 2006 The American Physical Society

but Λmax (σ λj ) is positive for the first 80 eigenvalues, i.e., all networks (if taken as fixed) would make the synchronization manifold unstable in at least 80 different transverse directions. Starting from random initial conditions, Figure 6.8 reports the temporal evolution of the synchronization error (t) =

N

|xi − x1 | + |yi − y1 | + |zi − z1 | 3(N − 1) j =2

at different values of the switching time τ , showing that the dynamical networked system is indeed able to synchronize, with a transient time Tsync to achieve synchronization (the time needed for δ(t) to become smaller than 0.1, shown in the inset) that scales with τ almost exponentially.

222


As opposed to slow commutative variations in the coupling wiring, Ref. [310] considered the case of mobile agents, each one associated with a chaotic oscillator coupled with the chaotic oscillators of the neighboring agents. In particular, Ref. [310] considered N moving individuals distributed in a planar space of size L with periodic boundary conditions. Each individual moves with velocity vi (t) and direction of motion θi (t) (v being the modulus of the agent velocity, which is the same for all individuals). In this model, the agents are random walkers that update stochastically their direction angle θi (t). As a consequence, the position and the orientation of each agent are updated according to: xi (t + tM ) = xi (t) + vi (t)tM θi (t + tM ) = ξi (t + tM ),

(6.30)

where xi (t) is the position of the ith agent in the plane at time t, and ξi (t) are N independent random variables chosen at each time with uniform probability in the interval [−π, π]. In addition, Ref. [310] considered the case that individuals may perform longdistance jumps. This is accounted for by defining a parameter pj that quantifies the probability for an individual to perform a jump into a completely random position. Ref. [310] studied the case in which each agent is carrying a dynamical system, characterized by a state variable vector yi (t) ∈ Rn which evolves according to the Rössler oscillator chaotic law. It is reasonable to assume that each agent interacts at a given time with only those agents located within a neighborhood of an interaction radius, defined as r. Under the hypothesis of a fast switching, that here corresponds to consider a time scale for the movement of the agents in the plane much faster than the time scale of the dynamics of the Rössler oscillator, Ref. [310] studied theoretically and numerically the emerging synchronization properties, and showed that the critical parameter for the arousal of a synchronous dynamics between the mobile oscillators is the density of the agents in the plane.

6.8. Synchronization and modular structures At the end of this survey on synchronization phenomena in complex networks, it is important to mention that synchronous motion in complex graphs gives the option of gathering information on the underlying topological structure of the connection wiring. Indeed, as we will see momentarily, the desynchronization scenario of a graph of coupled dynamical units helps to reveal the hierarchical modular structure that actually forms the skeleton of the underlying topology of connections.

6.8. Synchronization and modular structures

223

First of all, let us briefly introduce the concept of graph modules or community structures, as this property has been found to be ubiquitously present in natural and artificial networks. Modules or community structures in a graph are tightly connected subsets of nodes within which the network connections are dense, and between which connections are sparser. In other words, if one carefully observes the distribution of links in a network, one can picture out the graph as composed by a hierarchy of subgraphs (the network’s modules), each one of them grouping nodes that are densely connected among them, while connections are rare between nodes belonging to different modules. The importance of such a hierarchical structure is that the dynamical units belonging to each module arrange collectively to contribute to a given functioning of the network, and the collective functioning of the graph can be seen as the parallel action of each one of such modular tasks. The main problem encountered in the detection of the modular structure of a network is the large associated computational demand. Indeed, once the adjacency matrix of the whole network is given, the disentangling of each of the network’s modules is formally equivalent to the classical graph partitioning problem in computer science, which is known to be a NP-complete problem (i.e., the number of operations needed to fully reveal the hierarchical structure of network’s modules scales nonpolinomially with the network size). The associated computational demand has, therefore, constituted a big limitation to modules detection in large graphs. Recently, however, a very important study [311] revealed that different topological hierarchies can be associated to different dynamical time scales in the transient process leading the synchronization of networking oscillators. This observation has been later on elaborated, and a useful technique has been proposed [312] that combines topological and dynamical information on the graph in order to devise a Dynamical Clustering algorithm able to identify the modular structure of a graph in a fast and reliable way. We, indeed, have already discussed that an intermediate regime between global phase locking and full absence of synchronization in ensembles of coupled oscillators is represented by clusters of synchronized oscillators. This implies a division of the whole oscillators’ set (the graph) into groups of elements which oscillate at the same (average) frequency. The idea elaborated in Ref. [312] is that, starting from a fully synchronized state of a given network of coupled oscillators, a judiciously chosen dynamical change in the coupling of the interactions would yield a progressive hierarchical clustering of the oscillators, in which the clusters of synchronous motion would correspond to the modular structures of the graph.


224

In particular, Ref. [312] illustrated the idea with reference to a specific model, the so-called Opinion Changing Rate (OCR) model. This is a continuous-time system of coupled phase oscillators that was originally introduced for the modelling of opinion consensus in social networks [313], and represents a variation of the Kuramoto model discussed in [66, Section 7.6]. Let us start by assuming the knowledge of the adjacency matrix A = {aij } for a undirected, unweighted graph with N nodes and K edges. In order to reveal the full hierarchical modular structure, Ref. [312] artificially associated to each node i of the graph a dynamical variable xi (t) ∈ ]−∞, +∞[, and devised a dynamical model of the whole network given by:

α(t) σ bij sin(xj − xi )βe−β|xj −xi | . x˙i (t) = ωi + (6.31) α(t) j ∈Ni bij j ∈Ni In equations (6.31), ωi is the natural frequency of the ith oscillator in the network (in Ref. [312] the different ωi ’s were randomly selected from a uniform distribution between ωmin = 0 and ωmax = 1), σ is the coupling strength, and Ni is the set of nodes adjacent to i, i.e., all nodes j for which aij = aj i = 1. The other two important parameters of equations (6.31) are β and the set of bij . As for the parameter β, it has the effect of tuning the exponential factor in the coupling term of equations (6.31), i.e., it is responsible for switching off the interaction as soon as the phase distance between two oscillators exceeds a certain threshold (β = 3 was fixed in Ref. [312]). The set of bij are the node betweenness or load, that were already used for the weighting of the network interaction in equation (6.15). Here, instead, the interaction between two adjacent nodes i and j is weighted by a term

α(t) α(t) / bij , bij j ∈Ni

where α(t) is a time dependent exponent. The main idea of Ref. [312] is to start with α(t = 0) = 0, and to fix once forever the coupling strength σ to that value σ0 warranting that the unweighted (α = 0) initial graph displays a fully synchronized state. The next step is then to solve equations (6.31) with a progressively decreasing value of α(t). More precisely, Ref. [312] considered a stepwise process α(tl+1 ) = α(tl ) − δα for tl+1 > t > tl , where the instants at which the α parameter is updated are equi-spaced in time, i.e., tl+1 − tl = T

∀l.


225

Specifically, in Ref. [312] it was used T = 2. Let us pay attention now to the distribution of loads in the original graph. If the original graph is modular, this is reflected by a vastly inhomogeneous distribution of edge betweenness, in which the edges connecting nodes belonging to the same module (to two different modules) are associated to small (large) values of the betweenness. On its turn, this means that, as α starts to decrease from zero, the interaction on the edges connecting nodes belonging to the same module will be progressively strengthened, while the interaction on those edges connecting elements belonging to different modules will be progressively weakened. Now, as the network is originally prepared to be fully synchronized, it has to be expected that, as α decreases, each one of the network’s modules will stay arranged into a phase synchronized dynamics (all the links connecting elements of the module will correspond, indeed, to an increasing coupling strength), but the collective phase dynamics of one module will deviate from that of the other modules, as the coupling between elements of the modules will be progressively weakened. The expected scenario is, therefore, that the original full synchronization state will hierarchically split into clusters of synchronized elements, accordingly to the hierarchy of modules present in the graph, so that the individuation of the synchronization clusters will lead to the individuation of the network’s modules. Ref. [312] individuated the synchronization clusters in terms of those groups of nodes with the same instantaneous frequency x(t), ˙ and proposed to monitor the emerging set of synchronization clusters at each value of α(t), leading to have the full information on the hierarchical network division into modules. Furthermore, once the global picture of the network division is obtained, one can individuate the best modular division of the graph (i.e., the best α value) by looking at the maximum (as a function of α(t)) of the modularity parameter Q introduced in Ref. [246]. In order to comparatively evaluate the performance of the algorithm, Ref. [312] have considered, as in Ref. [246], a set of computer generated random graphs constructed in such a way to have a well defined modular structure. All graphs were generated with N = 128 nodes and K = 1,024 edges. The nodes were divided into four communities, each one of them containing 32 nodes, and the pairs of nodes belonging to the same module (to different modules) were linked with probability pin (pout ). In particular, the value of pout was selected so as the average number zout of edges a node forms with members of other communities could be controlled. Ref. [312] varied the values of zout , and selected pin so as to maintain a constant total average node degree = 16. Obviously, the progressive increase of zout is reflected by a weaker and weaker modular structure of the resulting network.

226


Figure 6.9. Fraction p of correctly identified nodes as a function of zout for computer generated graphs with N = 128 nodes, and an average degree = 16. Open and black circles refer to the results of the OCR and the OCR-HK models, respectively, while the results of the GN algorithm are reported with black squares [246], and those of the Newman Q-optimization fast algorithm are reported with stars [314]. The inset reports the scaling of the full CPU time needed to complete the network clusterization (in seconds) as a function of the number of nodes N . Reprinted with permission from Ref. [312]. © 2007 The American Physical Society

As a result, since the modular structure of the generated networks is directly imposed by the generation process, the accuracy of the identification method can be assessed by monitoring the fraction p of correctly classified nodes vs. the choice of zout . The results obtained in Ref. [312] are reported in Figure 6.9. Namely, the value of p (averaged over twenty different realizations of the computer generated graphs and of the initial conditions) is plotted as a function of zout , for the algorithm based on the OCR model of equations (6.31), with σ = 5.0 and δα = 0.1. One clearly appreciates that the resulting performance (open circles) is comparable to that of other methods based solely on the topology, such as the Girvan– Newmann GN method (black squares) [246] and the Newman Q-optimization fast algorithm (stars) [314].


227

Ref. [312] further demonstrated that the performance of the algorithm can be made even better by adding a simple modification to the model (6.31) which further stabilizes the system. The modification consists in changing in time the natural frequencies ω’s according to the idea of confidence bound introduced for the first time by Hegselmann and Krause (HK), in the context of models for opinion formation. Such a confidence bound is nothing but an extra parameter which fixes the range of compatibility of the nodes. Precisely, at each time, the generic node i, having a dynamical variable xi (t) and a natural frequency ωi (t), checks how many of its neighbors j are compatible, i.e., have a value of the variable xj (t) falling inside the confidence range [xi − , xi + ]. Then, at the following step in the numerical integration, one sets ωi (t + t), i.e., the node takes the average value of the ω’s of its compatible neighbors at time t. In other words, the OCR-HK superimposed changes of the ω(t)’s to the main dynamical evolution of equation (6.31), and contributes this way to stabilize the frequencies of the oscillators according to the correct modular structure of the network. The results obtained on the computer generated graphs, again for δα = 0.1, are reported in Figure 6.9 as black circles. Figure 6.9 furthermore allows for a comparison between the technique of individuating the network modules based on cluster desynchronization, and many other methods that only exploited information on the topology of the original graph. The main advantage of the technique proposed in Ref. [312] stands on its reduced computational demand [315]. For instance, the cluster desynhcronization technique needs to calculate the betweenness distribution only for the initial graph, whereas the high performance of iterative methods [246,314] needs to take into account the information on pathways redistribution all the times a given edge is removed. The scaling of the CPU time needed to perform the whole hierarchical scan of clusters is reported in the inset of Figure 6.9 versus the network size N , showing that, for sparse graphs of size up to N = 16,384, one has a scaling law of O(N 1.76 ) for the OCR-HK system. This implies that, as the calculation of edge betweenness takes O(N 2 ) operations, the improvement of this method over iterative ones becomes more and more evident as N increases, thus making this strategy particularly promising for the analysis of large size graphs.

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Subject Index

active–passive decomposition method, 14 adaptation, 213 almost synchronization, 3 amplitude turbulence, 160 analytic signal approach, 37 anti-phase synchronization, 146, 192 anticipating synchronization, 15, 179 – manifold, 15 Arnold tongue, 36, 43, 61 asymmetric coupling effects, 164 attractor bubbling, 20

convective Lyapunov exponent, 182 conventional synchronization, 10 correlation length, 164 coupling direction, 101 critical point, 211 cross-correlation function, 95 defect enhanced anomaly, 173 defect statistics, 153 degree distribution, 187 degree–degree correlation, 206 delayed dynamical systems, 83 deterministic and stochastic resonance, 129 differentiable generalized synchronization, 76 discrete coupling, 163 drive decomposable, 11 drive–response, 10 dynamical clustering, 223

Belousov–Zhabotinsky reaction, 174 Benjamin–Feir line, 160 bidirectional coupling, 10 – configuration, 4 binomial distribution, 105 characteristic path length, 188 circle maps, 136 cluster synchronization, 135 clustering coefficient, 189 coexisting attractors, 88 coherence resonance, 192 coherent chaotic oscillators, 58 combinatorial Laplacian matrix, 190 common noise, 113 community structures, 223 complete synchronization, 3, 10 complex Ginzburg–Landau equations, 157, 173 conditional Lyapunov exponents, 12, 13, 17, 21 conditional probability, 103, 188 configuration model, 206 connection graph stability method, 29 consistency, 112 constructive noise effects, 122 continuous extended systems, 157 contraction region, 116 convective instabilities, 178, 181

Eckhaus instability, 160 embedding space, 77, 96 false nearest neighbors, 96, 104 first order perturbation theory, 175 Fokker–Planck equation, 34 frequency synchronization, 167 generalized synchronization, 3, 32, 74 geodesic, 187 graph modules, 223 graph partitioning, 223 hard transition to synchronization, 137 Hilbert transform, 37, 102 homoclinic return, 117 homogeneous driving configuration, 13 identical synchronization, 3, 10 imperfect phase synchronization, 3, 32, 59 instantaneous frequency ratio, 103 instantaneous phase, 32 integral behavior, 164 241

242

Subject Index

intermittent lag synchronization, 3, 63 intermittent phase synchronization, 91 Jacobian matrix, 116 Kaplan–Yorke or Lyapunov dimension, 86 Kuramoto dynamics, 210 Kuramoto–Sivashinsky (KS) equations, 173 lag synchronization, 3, 32, 62 laminar periods, 47, 57 laminar phase, 57 Laplacian matrix, 189 load, 197 locally synchronous subsystems, 80 locking condition, 143 Lyapunov exponent, 69, 114, 194 Lyapunov function, 29 Lyapunov spectrum, 39, 64, 145 master stability function, 11, 20, 193, 215 master–slave coupling configuration, 10 mean conditional dispersion, 98 mean field, 192, 211 mixed state embedding, 100 modular structures, 222 mutation, 213 mutual false nearest neighbor, 98 mutual information, 95 mutual prediction error, 100 node betweenness, 188 node centrality, 188 noise enhanced phase synchronization, 120, 129 noise induced phase synchronization, 117, 124 noise-induced complete synchronization, 112, 126 noise-induced order, 114 noise-induced synchronization, 114 nonautonomous chaotic oscillators, 69 noncoherent oscillators, 59 on-off intermittencies, 93 opinion changing rate (OCR) model, 224 partial synchronization, 192 path, 187 Pearson statistic, 206

Pearson’s coefficient, 168 penetration depth, 154 period-doubling phase synchronization, 94 permanent synchronization domains, 156 phase, 3 – clustering, 192 – coherence, 32 – coherent, 4 – defects, 173 – entrainment, 41 – incoherent, 39 – locked states, 142 – slips, 34, 58, 121, 139, 170 – synchronization, 33, 47, 153, 167 – turbulence, 160 Poincaré section, 37, 40 preferred phase synchronized state, 91 propagation velocities, 149 propensity for synchronization, 197 ring intermittency, 56 rotation number, 137 σ -synchronization, 83 saddle-node bifurcation, 55 scale free networks, 198, 205 Shannon entropy, 95, 103, 119 shortest path, 187 similarity function, 62 small world networks, 199 small world property, 188 soft transition, 137 stochastic resonance, 122, 133 strong synchronization, 76 symmetric coupling, 26 sympathy, 2 synchrogram, 104 synchronizability, 25, 27, 195 synchronization, 3 – domains, 146, 151 – error, 12 – function, 80 – index, 103 – manifold, 10, 18, 22, 219 – points, 81 – – percentage, 104, 107 synchronizing signal, 12 target degree, 207 transversal Lyapunov exponents, 18

Subject Index type-I intermittency, 47 type-I intermittent, 51

unstable manifold, 124 unstable periodic orbit, 43, 51

unidirectional, 4 – coupling, 10 – – configuration, 202

walk, 187 weak synchronization, 76 weighted graph, 190

243

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