Differential Geometry Applied to Dynamical Systems (World Scientific Series on Nonlinear Science, Series a)

DIFFERENTIAL GEOMETRY APPLIED TO DYNAMICAL SYSTEMS WORLD SCIENTIFIC SERIES ON NONLINEAR SCIENCE Editor: Leon O. Chua ...

Author: Jean-marc Ginoux

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DIFFERENTIAL GEOMETRY APPLIED TO DYNAMICAL SYSTEMS

WORLD SCIENTIFIC SERIES ON NONLINEAR SCIENCE Editor: Leon O. Chua University of California, Berkeley Series A.

MONOGRAPHS AND TREATISES*

Volume 48:

Bio-Inspired Emergent Control of Locomotion Systems M. Frasca, P. Arena & L. Fortuna

Volume 49:

Nonlinear and Parametric Phenomena V. Damgov

Volume 50:

Cellular Neural Networks, Multi-Scroll Chaos and Synchronization M. E. Yalcin, J. A. K. Suykens & J. P. L. Vandewalle

Volume 51:

Symmetry and Complexity K. Mainzer

Volume 52:

Applied Nonlinear Time Series Analysis M. Small

Volume 53:

Bifurcation Theory and Applications T. Ma & S. Wang

Volume 54:

Dynamics of Crowd-Minds A. Adamatzky

Volume 55:

Control of Homoclinic Chaos by Weak Periodic Perturbations R. Chacón

Volume 56:

Strange Nonchaotic Attractors U. Feudel, S. Kuznetsov & A. Pikovsky

Volume 57:

A Nonlinear Dynamics Perspective of Wolfram's New Kind of Science L. O. Chua

Volume 58:

New Methods for Chaotic Dynamics N. A. Magnitskii & S. V. Sidorov

Volume 59:

Equations of Phase-Locked Loops J. Kudrewicz & S. Wasowicz

Volume 60:

Smooth and Nonsmooth High Dimensional Chaos and the Melnikov-Type Methods J. Awrejcewicz & M. M. Holicke

Volume 61:

A Gallery of Chua Attractors (with CD-ROM) E. Bilotta & P. Pantano

Volume 62:

Numerical Simulation of Waves and Fronts in Inhomogeneous Solids A. Berezovski, J. Engelbrecht & G. A. Maugin

Volume 63:

Advanced Topics on Cellular Self-Organizing Nets and Chaotic Nonlinear Dynamics to Model and Control Complex Systems R. Caponetto, L. Fortuna & M. Frasca

Volume 64:

Control of Chaos in Nonlinear Circuits and Systems B. W.-K. Ling, H. H-C. Lu & H.K. Lam

Volume 65:

Chua’s Circuit Implementations: Yesterday, Today and Tomorrow L. Fortuna, M. Frasca & M. G. Xibilia

*

To view the complete list of the published volumes in the series, please visit: http://www.worldscibooks.com/series/wssnsa_series.shtml

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NONLINEAR SCIENCE WORLD SCIENTIFIC SERIES ON

Series A

Vol. 66

Series Editor: Leon O. Chua

DIFFERENTIAL GEOMETRY APPLIED TO DYNAMICAL SYSTEMS

Jean-Marc Ginoux Université du Sud, France

World Scientific NEW JERSEY

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LONDON

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SINGAPORE

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BEIJING

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SHANGHAI

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Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

World Scientific Series on Nonlinear Science, Series A — Vol. 66 DIFFERENTIAL GEOMETRY APPLIED TO DYNAMICAL SYSTEMS (With CD-ROM) Copyright © 2009 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

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ISBN-13 978-981-4277-14-3 ISBN-10 981-4277-14-2

Printed in Singapore.

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Differential Geometry Applied to Dynamical Systems

“. . . every time the system absorbs energy the curvature of its trajectory decreases and viceversa . . . ”. — N. Minorsky1

1 N.

Minorsky (1967, p. 108).

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Preface

The study of dynamical systems, i.e. of systems of differential equations finds roots in the works of L. A. Cauchy (1835) and the so-called calculus of limits which gave rise to an analytic approach1. Thus, many methods based on regular expansions enabled to deduce local behaviors of dynamical systems. Then, a geometric approach was initiated by Henri Poincaré (18811886) in his famous memoirs: Sur les courbes définies par une équation différentielle which represent the foundations of the qualitative 2 or geometric theory of differential equations. Continued during the XXth century with the works of G. Valiron (1950), S. Lefschetz (1957), V. V. Nemytskii & V. V. Stepanov (1960), N. Minorsky (1962-1967), F. Brauer & J. A. Nohel (1969), . . . it seems nevertheless that Differential Geometry had been rarely used for dynamical systems3 study. The aim of this book is to present a new approach which consists of applying Differential Geometry to Dynamical Systems and is called Flow Curvature Method . Thus, while considering the trajectory curve, integral of any n-dimensional dynamical system, as a curve in Euclidean n-space, the curvature of the trajectory curve, i.e. curvature of the flow may be analytically computed. Then, the location of the points where the curvature of the flow vanishes defines a manifold called: flow curvature manifold . It will be stated that, since such a manifold is defined starting from the 1 See

J. Molk (1910) and E. L. Ince (1926) p. 529 for a History of differential equations. C. Gilain (1977) for details about Poincaré’s geometric approach of differential equations. 3 The oldest reference which has been indicated to me by Prof. C. Mira is: M. Haag (1879). 2 See

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time derivatives of the velocity vector field and so, contains information about the dynamics of the system, its only knowledge enables to find again the main features of the dynamical system studied. These features may be considered as the foundations of Dynamical Systems Theory. There are six of them: differential equations, dynamical systems, invariant sets, local bifurcations, slow-fast dynamical systems, integrability and to each of these concepts corresponds a chapter. Thus, this manuscript has been designed in a symmetric manner and consists of three parts each of them comprising these six chapters. The first part which may be regarded independently of the two others is a detailed presentation of these six chapters from the analytic point of view of Dynamical Systems Theory accompanied by references4 , anecdotes and many examples. Chapter 1, Introduction, is an historical presentation of differential equations used to modelize natural phenomena. In Ch. 2, Dynamical systems, state space and flow definitions, existence and uniqueness and Liapounoff stability theorems are summarized and emphasized with original references and significant examples as well as the notion of Poincaré index , the concept of limit cycle or strange attractor . Then, definitions of first integral and Lie derivative which will be extensively used in this book are presented. Hamiltonian integrable systems and K.A.M. theorem are also recalled. Chapter 3, Invariant sets, consists of definitions of global (resp. local) invariant manifolds and stable manifold theorem for a fixed point. Chapter 4 entitled local bifurcations is devoted to the Center Manifold Theorem and Normal Form Theorem which are presented with original proofs and highlighted through examples as well as local bifurcations such as saddle-node, transcritical, pitchfork or Hopf bifurcations. In Ch. 5, Slow-Fast Dynamical Systems, definitions of singularly perturbed dynamical systems and slow-fast dynamical systems are proposed. Then, the so-called Geometric Singular Perturbation Theory and the concept of slow invariant manifold are recalled and emphasized with paradigmatic Van der Pol and Chua systems. In Ch. 6, Integrability, integrability conditions, integrating factor and multiplier of dynamical systems are reminded. Then, Darboux Theory of Integrability is presented for the first time with its original proofs and examples applied to dynamical systems.

4 Historical

references to original works are made by page.

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The second part is exactly symmetric5 to the first one since it involves the same chapters and concepts as those previously defined but then considered from the Differential Geometry point of view. Chapter 7, Differential Geometry, is a presentation of the concepts inherent to Differential Geometry such as curves, osculating plane and curvatures. By considering the trajectory curves integral of any n-dimensional dynamical systems as curves in Euclidean n-space which possess local metrics properties of curvatures enables to define a manifold called: flow curvature manifold. Let’s note that the point of view is completely different from the previous one since it deals with curvature of trajectory curves instead of vector field, i.e. one substitutes a manifold to a differential equation, to a dynamical system. Thus, the Flow Curvature Method is based on the idea that if it is generally impossible to have a closed form of the trajectory curve it is still possible to analytically compute its curvature since it only involves its time derivatives. Then, it will be stated in chapters 8, 9, 10 & 11 that all the results found in chapters 2, 3, 4, 5 & 6 such as fixed points stability, invariant sets, centre manifold , normal forms, local bifurcations, slow invariant manifold and integrability of dynamical systems may be found again according to the Flow Curvature Method , i.e. starting from the flow curvature manifold. In Ch. 8, it is stated that the Flow Curvature Method enables to find again stability theorems for fixed points of low-dimensional two and three dynamical systems according to a theorem due to Henri Poincaré. In Ch. 9, concepts of global invariance and local invariance, which are of great importance since all the proofs are based on them, are (re)defined from Darboux invariance theorem. Then, it will be stated that flow curvature manifold also enables to “detect” linear invariant manifolds of any n-dimensional dynamical systems which may be used to build first integrals of these systems. For nonlinear invariant manifolds identity between flow curvature manifold and the so-called extatic manifolds is also stated. In Ch. 10, it is established that the Flow Curvature Method enables to easily compute the coefficients of the centre manifold approximation of any n-dimensional dynamical systems according to global invariance of the 5 Chapters

of part two (three) are chapters of part one (two) incremented of 6.

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flow curvature manifold. Then, a link between normal forms of dynamical systems and “normal forms” of flow curvature manifold will be highlighted. Such a link enables to directly compute the normal form of a dynamical system starting from its flow curvature manifold. In Ch. 11, by considering singularly perturbed systems comprising a small multiplicative parameter ε in factor in their velocity vector field, identity between Geometric Singular Perturbation Theory and Flow Curvature Method is pointed out up to suitable order in ε. Moreover, identity between Fenichel’s invariance and Darboux invariance theorem is demonstrated. Then, it is stated that the 1st flow curvature manifold associated with a two-dimensional dynamical system directly provides a first order approximation in ε of the slow invariant manifold given by Geometric Singular Perturbation Theory while the 2nd flow curvature manifold associated with a three-dimensional dynamical system directly provides a second order approximation in ε of the slow invariant manifold . High orders approximation of the slow invariant manifold may be simply obtained by replacing the flow curvature manifold by its successive Lie derivatives. The main difference between Flow Curvature Method and the so-called Geometric Singular Perturbation Theory is that flow curvature manifold directly provides the slow invariant manifold analytical equation of any ndimensional slow-fast dynamical systems not only singularly perturbed but also for non-singularly perturbed as exemplified with Lorenz model. Invariance of the flow curvature manifold, i.e. of the slow manifold is then stated according to Darboux invariance theorem. In Ch. 12, Darboux theory of integrability is conjugated with Flow Curvature Method in order to build first integrals of dynamical systems. Many examples of two and three-dimensional dynamical systems such as VolterraLotka, Kapteyn-Bautin, . . . enable to highlight the efficiency of Flow Curvature Method for integrability. In Ch. 13, Inverse problem, while considering that the only knowledge about a polynomial dynamical system is its flow curvature manifold, it is stated that one may find a family of vector field comprising this polynomial dynamical system solving thus the inverse problem.

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The third part of this book consists of applications of Flow Curvature Method to all these concepts. In Chapter 14, Flow Curvature Method enables to find again fixed points stability of FitzHugh-Nagumo and PikovskiiRabinovich-Trakhtengerts (PRT) two and three dimensional dynamical systems. In Ch. 15, Flow Curvature Method enables to detect invariant manifolds of Pikovskii-Rabinovich-Trakhtengerts, Rikitake, Chua and Lorenz three-dimensional dynamical systems and are then used in order to build first integrals of these systems. In Ch. 16, Flow Curvature Method directly provides centre manifolds of Chua and Lorenz models and so highlights local bifurcations. In Ch. 17, Flow Curvature Method directly provides the slow invariant manifold of many n-dimensional dynamical systems such as: • piecewise linear models of dimensions two and three (Van der Pol, Chua), dimensions four and five (Chua), • singularly perturbed systems of dimensions two and three (FitzHughNagumo, Chua), dimensions four and five (Chua), • slow-fast systems of dimensions two and three (Brusselator, (PRT), Rikitake), dimensions four and five (Homopolar dynamo, Mofatt, magnetoconvection). At last, forced Van der Pol system is used in order to show that Flow Curvature Method may be extended to the study of non-autonomous dynamical systems and more particularly for the computation of their slow invariant manifold analytical equation. In Appendix, many concepts inherent to Differential Geometry used in this book are recalled and identities necessary to the establishment of proofs are stated. Then, a generalization up to dimension n of the Tangent Linear System Approximation introduced by Rossetto et al. (1998) in order to obtain the slow manifold of slow-fast dynamical systems starting from the eigenvectors associated with the functional jacobian matrix is also presented.

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Since all the main features of Dynamical Systems Theory may be found again according to the Flow Curvature Method , i.e. starting from the flow curvature manifold both Dynamical Systems Theory and Flow Curvature Method are consistent and so Flow Curvature Method represents an alternative geometric approach for the study of dynamical systems which may be applied to autonomous as well as non-autonomous n-dimensional dynamical systems. The main results provided by the Flow Curvature Method are summarized in synopsis below. Each topic may be followed along the book by adding 6, e.g., Invariant Sets corresponds to Chapters 3, 6 and 9. All the examples used in the first part of this book through the point of view of Dynamical Systems Theory are then considered according to the framework of Differential Geometry. Moreover, the Mathematica files MF XX with which they have been elaborated are available on the included CD and also at: http://ginoux.univ-tln.fr J. M. Ginoux

Dynamical Systems Ch. 2, 8, 14

Invariant Sets Ch. 3, 9, 15

Local Bifurcations Ch. 4, 10, 16

Flow Curvature Method

SlowFast Systems Ch. 5, 11, 17

Integrability Ch. 6, 12, 15

Fig. 1

Synopsis

Inverse Problem Ch. 13

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Acknowledgments

This manuscript is the result of a multitude of encounters and discussions with many searchers and scientists who gave me through their advice the necessary impulsion to achieve my research that’s the reason why I would like to pay homage to them: Professors Dana Schlomiuk, Claudine Delcarte, Bruno Cessac, Alain Pumir, Christian Mira, Jean-Marie Strelcyn, Jaume Llibre, Cyrille Bertelle, Michel Cotsfatis, Gérard Duchamps, Robert Gilmore, Bernd Krauskopf, Eric Benoˆıt, Jean-Pierre Fran¸coise, Pierre Auger, Alain Goriély, Didier Sornette, J¨ urgen Kurths, Riccardo Meucci . . . to name but a few. I would like also to address special thanks to Professors Aziz-Alaoui, Christophe Letellier, Jean-Louis Jamet, René Lozi and Leon O. Chua who have been present at all steps of this work and of course to Bruno Rossetto who helped me and encouraged me to continue these research these past few years. To my family and my wife who have supported me in this work I would like to extend my gratitude and love.

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Contents

Preface

vii

Acknowledgments

xiii

List of Figures

xxiii

List of Examples

xxv

Dynamical Systems

1

1. Differential Equations

3

1.1 1.2 1.3

Galileo’s pendulum . . . . . . . . . . . . . . . . . . . . . . D’Alembert transformation . . . . . . . . . . . . . . . . . From differential equations to dynamical systems . . . . .

2. Dynamical Systems 2.1 2.2 2.3 2.4 2.5

2.6

3 5 6 7

State space – phase space . . . . . . . . . . . . . Definition . . . . . . . . . . . . . . . . . . . . . . Existence and uniqueness . . . . . . . . . . . . . Flow, fixed points and null-clines . . . . . . . . . Stability theorems . . . . . . . . . . . . . . . . . 2.5.1 Linearized system . . . . . . . . . . . . . 2.5.2 Hartman-Grobman linearization theorem 2.5.3 Liapounoff stability theorem . . . . . . . Phase portraits of dynamical systems . . . . . . . 2.6.1 Two-dimensional systems . . . . . . . . . 2.6.2 Three-dimensional systems . . . . . . . . xv

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2.7

2.8

2.9

2.10

Various types of dynamical systems . . . . . . . . 2.7.1 Linear and nonlinear dynamical systems 2.7.2 Homogeneous dynamical systems . . . . 2.7.3 Polynomial dynamical systems . . . . . . 2.7.4 Singularly perturbed systems . . . . . . . 2.7.5 Slow-Fast dynamical systems . . . . . . . Two-dimensional dynamical systems . . . . . . . 2.8.1 Poincaré index . . . . . . . . . . . . . . . 2.8.2 Poincaré contact theory . . . . . . . . . . 2.8.3 Poincaré limit cycle . . . . . . . . . . . . 2.8.4 Poincaré-Bendixson Theorem . . . . . . . High-dimensional dynamical systems . . . . . . . 2.9.1 Attractors . . . . . . . . . . . . . . . . . 2.9.2 Strange attractors . . . . . . . . . . . . . 2.9.3 First integrals and Lie derivative . . . . . Hamiltonian and integrable systems . . . . . . . 2.10.1 Hamiltonian dynamical systems . . . . . 2.10.2 Integrable system . . . . . . . . . . . . . 2.10.3 K.A.M. Theorem . . . . . . . . . . . . .

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3. Invariant Sets 3.1

3.2

Manifold . . . . . . . . . . 3.1.1 Definition . . . . . 3.1.2 Existence . . . . . Invariant sets . . . . . . . 3.2.1 Global invariance 3.2.2 Local invariance .

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4. Local Bifurcations 4.1

4.2 4.3

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Center Manifold Theorem . . . . . . . . . . . . . . . . 4.1.1 Center manifold theorem for flows . . . . . . . 4.1.2 Center manifold approximation . . . . . . . . 4.1.3 Center manifold depending upon a parameter Normal Form Theorem . . . . . . . . . . . . . . . . . . Local Bifurcations of Codimension 1 . . . . . . . . . . 4.3.1 Saddle-node bifurcation . . . . . . . . . . . . . 4.3.2 Transcritical bifurcation . . . . . . . . . . . . 4.3.3 Pitchfork bifurcation . . . . . . . . . . . . . .

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4.3.4

Hopf bifurcation

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5. Slow-Fast Dynamical Systems 5.1 5.2

5.3

69

Introduction . . . . . . . . . . . . . . . . . . . . . . . Geometric Singular Perturbation Theory . . . . . . . 5.2.1 Assumptions . . . . . . . . . . . . . . . . . . 5.2.2 Invariance . . . . . . . . . . . . . . . . . . . 5.2.3 Slow invariant manifold . . . . . . . . . . . . Slow-fast dynamical systems – Singularly perturbed systems . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Singularly perturbed systems . . . . . . . . . 5.3.2 Slow-fast autonomous dynamical systems . .

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6. Integrability 6.1

6.2

6.3

85

Integrability conditions, integrating factor, multiplier . . 6.1.1 Two-dimensional dynamical systems . . . . . . . 6.1.2 Three-dimensional dynamical systems . . . . . . First integrals – Jacobi’s last multiplier theorem . . . . 6.2.1 First integrals . . . . . . . . . . . . . . . . . . . 6.2.2 Jacobi’s last multiplier theorem . . . . . . . . . Darboux theory of integrability . . . . . . . . . . . . . . 6.3.1 Algebraic particular integral – General integral 6.3.2 General integral . . . . . . . . . . . . . . . . . . 6.3.3 Multiplier . . . . . . . . . . . . . . . . . . . . . 6.3.4 Algebraic particular integral and fixed points . . 6.3.5 Homogeneous polynomial dynamical systems of degree m . . . . . . . . . . . . . . . . . . . . . . 6.3.6 Homogeneous polynomial dynamical systems of degree two . . . . . . . . . . . . . . . . . . . . . 6.3.7 Planar polynomial dynamical systems . . . . . .

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Differential Geometry

121

7. Differential Geometry 7.1

Concept of curves – Kinematics vector functions 7.1.1 Trajectory curve . . . . . . . . . . . . . . 7.1.2 Instantaneous velocity vector . . . . . . . 7.1.3 Instantaneous acceleration vector . . . .

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7.2

7.3 7.4

7.5

Gram-Schmidt process – Generalized Frénet moving frame 7.2.1 Gram-Schmidt process . . . . . . . . . . . . . . . 7.2.2 Generalized Frénet moving frame . . . . . . . . . Curvatures of trajectory curves – Osculating planes . . . . Curvatures and osculating plane of space curves . . . . . . 7.4.1 Frénet trihedron – Serret-Frénet formulae . . . . . 7.4.2 Osculating plane . . . . . . . . . . . . . . . . . . . 7.4.3 Curvatures of space curves . . . . . . . . . . . . . Flow curvature method . . . . . . . . . . . . . . . . . . . 7.5.1 Flow curvature manifold . . . . . . . . . . . . . . 7.5.2 Flow curvature method . . . . . . . . . . . . . . .

8. Dynamical Systems 8.1

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Phase portraits of dynamical systems . . . . . . . . . . . . 135 8.1.1 Fixed points . . . . . . . . . . . . . . . . . . . . . 135 8.1.2 Stability theorems . . . . . . . . . . . . . . . . . . 137

9. Invariant Sets 9.1

9.2 9.3

Invariant manifolds . . . . . . 9.1.1 Global invariance . . 9.1.2 Local invariance . . . Linear invariant manifolds . . Nonlinear invariant manifolds

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10. Local Bifurcations 10.1

10.2 10.3

Center Manifold . . . . . . . . . . . . . 10.1.1 Center manifold approximation 10.1.2 Center manifold depending upon Normal Form Theorem . . . . . . . . . . Local bifurcations of codimension 1 . . .

11.2 11.3

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11. Slow-Fast Dynamical Systems 11.1

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Slow manifold of n-dimensional slow-fast dynamical systems . . . . . . . . . . . . . . . . . . . . . . . . . Invariance . . . . . . . . . . . . . . . . . . . . . . . . Flow Curvature Method – Singular Perturbation Method . . . . . . . . . . . . . . . . . . . . . . . . . 11.3.1 Darboux invariance – Fenichel’s invariance .

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11.4

xix

11.3.2 Slow invariant manifold . . . . . . . . . . . . . . . 191 Non-singularly perturbed systems . . . . . . . . . . . . . . 200

12. Integrability 12.1

12.2 12.3

203

First integral . . . . . . . . . . . . . . . . . . . . . . . 12.1.1 Global first integral . . . . . . . . . . . . . . . 12.1.2 Local first integral . . . . . . . . . . . . . . . . Linear invariant manifolds as first integral . . . . . . . Darboux theory of integrability . . . . . . . . . . . . . 12.3.1 General integral – Multiplier . . . . . . . . . . 12.3.2 Darboux homogeneous polynomial dynamical systems of degree two . . . . . . . . . . . . . . 12.3.3 Planar polynomial dynamical systems . . . . .

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13. Inverse Problem 13.1

13.2

13.3

215

Flow curvature manifold of polynomial dynamical systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.1.1 Two-dimensional polynomial dynamical systems . 13.1.2 Three-dimensional polynomial dynamical systems Flow curvature manifold symmetry (parity) . . . . . . . . 13.2.1 Two-dimensional polynomial dynamical systems . 13.2.2 n-dimensional polynomial dynamical systems . . . Inverse problem for polynomial dynamical systems . . . . 13.3.1 Two-dimensional polynomial dynamical systems . 13.3.2 Three-dimensional polynomial dynamical systems

Applications

227

FitzHugh-Nagumo model . . . . . . . . . . . . . . . . . . 227 Pikovskii-Rabinovich-Trakhtengerts model . . . . . . . . . 228

15. Invariant Sets - Integrability 15.1 15.2 15.3 15.4

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14. Dynamical Systems 14.1 14.2

203 203 204 206 209 209

Pikovskii-Rabinovich-Trakhtengerts model Rikitake model . . . . . . . . . . . . . . . Chua’s model . . . . . . . . . . . . . . . . Lorenz model . . . . . . . . . . . . . . . .

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16. Local Bifurcations 16.1 16.2

237

Chua’s model . . . . . . . . . . . . . . . . . . . . . . . . . 237 Lorenz model . . . . . . . . . . . . . . . . . . . . . . . . . 239

17. Slow-Fast Dynamical Systems – Singularly Perturbed Systems 17.1

17.2

17.3

17.4

17.5

17.6

17.7 17.8

Piecewise Linear Models 2D & 3D . . . . . . . . . 17.1.1 Van der Pol piecewise linear model . . . . 17.1.2 Chua’s piecewise linear model . . . . . . . Singularly Perturbed Systems 2D & 3D . . . . . . 17.2.1 FitzHugh-Nagumo model . . . . . . . . . . 17.2.2 Chua’s model . . . . . . . . . . . . . . . . Slow Fast Dynamical Systems 2D & 3D . . . . . . 17.3.1 Brusselator model . . . . . . . . . . . . . . 17.3.2 Pikovskii-Rabinovich-Trakhtengerts model 17.3.3 Rikitake model . . . . . . . . . . . . . . . . Piecewise Linear Models 4D & 5D . . . . . . . . . 17.4.1 Chua’s fourth-order piecewise linear model 17.4.2 Chua’s fifth-order piecewise linear model . Singularly Perturbed Systems 4D & 5D . . . . . . 17.5.1 Chua’s fourth-order cubic model . . . . . . 17.5.2 Chua’s fifth-order cubic model . . . . . . . Slow Fast Dynamical Systems 4D & 5D . . . . . . 17.6.1 Homopolar dynamo model . . . . . . . . . 17.6.2 Mofatt model . . . . . . . . . . . . . . . . 17.6.3 Magnetoconvection model . . . . . . . . . Slow manifold gallery . . . . . . . . . . . . . . . . Forced Van der Pol model . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . .

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241 . . . . . . . . . . . . . . . . . . . . . .

241 241 243 245 245 247 248 248 249 250 251 251 253 255 255 257 258 258 260 261 263 263

Discussion

265

Appendix A

269

A.1 A.2 A.3 A.4 A.5

269 270 270 271 272 273

Lie derivative . . . . . . . . . . . . . . . . . . . . . . . . . Hessian . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jordan form . . . . . . . . . . . . . . . . . . . . . . . . . . Connected region . . . . . . . . . . . . . . . . . . . . . . . Fractal dimension . . . . . . . . . . . . . . . . . . . . . . . A.5.1 Kolmogorov or capacity dimension . . . . . . . . . A.5.2 Liapounoff exponents – Wolf, Swinney, Vastano algorithm . . . . . . . . . . . . . . . . . . . . . . . .

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Contents

A.5.3 A.5.4 A.6

A.7

A.8

A.9

Liapounoff dimension and Kaplan-Yorke conjecture Liapounoff dimension and Chlouverakis-Sprott conjecture . . . . . . . . . . . . . . . . . . . . . . Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . A.6.1 Concept of curves . . . . . . . . . . . . . . . . . . A.6.2 Gram-Schmidt process and Frénet moving frame . A.6.3 Frénet trihedron and curvatures of space curves . A.6.4 First identity . . . . . . . . . . . . . . . . . . . . . A.6.5 Second identity . . . . . . . . . . . . . . . . . . . A.6.6 Third identity . . . . . . . . . . . . . . . . . . . . Homeomorphism and diffeomorphism . . . . . . . . . . . . A.7.1 Homeomorphism . . . . . . . . . . . . . . . . . . . A.7.2 Diffeomorphism . . . . . . . . . . . . . . . . . . . Differential equations . . . . . . . . . . . . . . . . . . . . A.8.1 Two-dimensional dynamical systems . . . . . . . . A.8.2 Three-dimensional dynamical systems . . . . . . . Generalized Tangent Linear System Approximation . . . . A.9.1 Assumptions . . . . . . . . . . . . . . . . . . . . . A.9.2 Corollaries . . . . . . . . . . . . . . . . . . . . . .

DGeometry

xxi

274 275 276 276 277 279 280 281 282 283 283 283 283 283 284 285 285 285

Mathematica Files

291

Bibliography

297

Index

309

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List of Figures

1

Synopsis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xii

1.1

Galileo’s pendulum . . . . . . . . . . . . . . . . . . . . . . . . .

4

2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10 2.11 2.12

Free fall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Volterra-Lotka predator-prey model . . . . . . . . . . . . . . . Phase plane stability diagram . . . . . . . . . . . . . . . . . . . Inverted pendulum . . . . . . . . . . . . . . . . . . . . . . . . . Stability diagram . . . . . . . . . . . . . . . . . . . . . . . . . . Saddle-focus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Poincaré limit cycle . . . . . . . . . . . . . . . . . . . . . . . . Duffing oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . Lorenz butterfly . . . . . . . . . . . . . . . . . . . . . . . . . . Spherical pendulum . . . . . . . . . . . . . . . . . . . . . . . . Hénon-Heiles Hamiltonian . . . . . . . . . . . . . . . . . . . . . Transversal Poincaré section (p2 , q2 ) of Hénon-Heiles Hamiltonian

10 12 16 17 20 21 28 30 33 36 38 39

3.1

Stable W S and unstable W U manifolds . . . . . . . . . . . . .

46

4.1

Part of the center manifold in green . . . . . . . . . . . . . . .

49

6.1

General integral . . . . . . . . . . . . . . . . . . . . . . . . . .

95

7.1

Osculating plane . . . . . . . . . . . . . . . . . . . . . . . . . . 130

8.1

Duffing oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . 142

9.1

Local invariance . . . . . . . . . . . . . . . . . . . . . . . . . . 148 xxiii

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10.1 Center manifold . . . . . . . . . . . . . . . . . . . . . . . . . . 164 11.1 Van der Pol slow invariant manifold . . . . . . . . . . . . . . . 195 11.2 Chua’s slow invariant manifold in (xz)-plane . . . . . . . . . . 199 11.3 Lorenz slow invariant manifold . . . . . . . . . . . . . . . . . . 201 12.1 Local first integral of Van der Pol model . . . . . . . . . . . . . 205 12.2 Volterra-Lotka’s first integral . . . . . . . . . . . . . . . . . . . 209 12.3 First integral of quadratic system . . . . . . . . . . . . . . . . . 211 17.1 Van der Pol piecewise linear model slow invariant manifold . . 17.2 Chua’s piecewise linear model slow invariant manifold . . . . . 17.3 FitzHugh-Nagumo model slow invariant manifold . . . . . . . 17.4 Chua’s cubic model slow invariant manifold . . . . . . . . . . . 17.5 Brusselator’s model slow invariant manifold . . . . . . . . . . . 17.6 (PRT) model slow invariant manifold . . . . . . . . . . . . . . 17.7 Rikitake model slow invariant manifold . . . . . . . . . . . . . 17.8 Chua’s model invariant hyperplanes in (x1 , x2 , x3 ) phase space 17.9 Chua’s model invariant hyperplanes in (x1 , x2 , x3 ) phase space 17.10 Chua’s model slow invariant manifold . . . . . . . . . . . . . . 17.11 Chua’s slow invariant manifold . . . . . . . . . . . . . . . . . . 17.12 Dynamo model slow invariant manifold . . . . . . . . . . . . . 17.13 Mofatt model slow invariant manifold . . . . . . . . . . . . . . 17.14 Magnetoconvection slow invariant manifold . . . . . . . . . . 17.15 (a) Chemical kinetics model. (b) Neuronal bursting model. . . 17.16 Forced Van der Pol model slow invariant manifold . . . . . . . Chua’s cubic model attractor structure . . . . . . . . . . . . .

243 245 246 247 248 249 250 253 255 256 258 259 261 262 263 264 267

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List of Examples

1.1. Order and degree of differential equations. . . 2.1. Free fall. . . . . . . . . . . . . . . . . . . . . . 2.2. Volterra-Lotka predator-prey model. . . . . . 2.3. Inverted pendulum. . . . . . . . . . . . . . . . 2.4. Saddle-focus. . . . . . . . . . . . . . . . . . . . 2.5. Linear and nonlinear systems. . . . . . . . . . 2.6. Quadratic systems. . . . . . . . . . . . . . . . 2.7. Slow-fast dynamical system. . . . . . . . . . . 2.8. Van der Pol model. . . . . . . . . . . . . . . . 2.9. Historically first limit cycle. . . . . . . . . . . 2.10. Duffing oscillator. . . . . . . . . . . . . . . . 2.11. Lorenz butterfly. . . . . . . . . . . . . . . . . 2.12. Harmonic oscillator. . . . . . . . . . . . . . . 2.13. Spherical pendulum. . . . . . . . . . . . . . . 2.14. Hénon-Heiles Hamiltonian. . . . . . . . . . . 3.1. Global invariant sets. . . . . . . . . . . . . . . 3.2. Stable and unstable manifolds. . . . . . . . . . 4.1. Nonuniqueness of the center manifold. . . . . 4.2. Center manifold 2D. . . . . . . . . . . . . . . 4.3. Center manifold 3D. . . . . . . . . . . . . . . 4.4. Center manifold depending upon a parameter. 4.5. Normal forms. . . . . . . . . . . . . . . . . . . 4.6. Normal forms. . . . . . . . . . . . . . . . . . . 4.7. Saddle-node bifurcation. . . . . . . . . . . . . 4.8. Transcritical bifurcation. . . . . . . . . . . . . 4.9. Pitchfork bifurcation. . . . . . . . . . . . . . . 4.10. Hopf bifurcation. . . . . . . . . . . . . . . . . xxv

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4 9 11 17 21 22 23 23 26 27 30 33 34 35 38 42 45 48 50 51 53 57 59 63 64 66 67

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5.1. Paradigm of Relaxation Oscillator: Van der Pol model. 5.2. Paradigm of Complex Dynamics. . . . . . . . . . . . . 5.3. Van der Pol system . . . . . . . . . . . . . . . . . . . . 5.4. Chua’s system . . . . . . . . . . . . . . . . . . . . . . . 5.5. Lorenz system . . . . . . . . . . . . . . . . . . . . . . . 6.1. Integrating factor. . . . . . . . . . . . . . . . . . . . . . 6.2. Integrable system. . . . . . . . . . . . . . . . . . . . . . 6.3. Multiplier. . . . . . . . . . . . . . . . . . . . . . . . . . 6.4. Integrable system. . . . . . . . . . . . . . . . . . . . . . 6.5. Integrable system. . . . . . . . . . . . . . . . . . . . . . 6.6. Darboux invariance theorem. . . . . . . . . . . . . . . . 6.7. General integral. . . . . . . . . . . . . . . . . . . . . . . 6.8. Muliplier. . . . . . . . . . . . . . . . . . . . . . . . . . 6.9. General integral of homogeneous dynamical systems. . 6.10. Multiplier of homogeneous dynamical systems. . . . . 6.11. Particular integral and fixed points. . . . . . . . . . . 6.12. General integral of Volterra-Lotka system. . . . . . . . 6.13. First integral of affine and projective models. . . . . . 6.14. Kapteyn-Bautin system. . . . . . . . . . . . . . . . . . 8.1. Flow curvature manifold and fixed points. . . . . . . . 8.2. Duffing oscillator. . . . . . . . . . . . . . . . . . . . . . 8.3. Lorenz model. . . . . . . . . . . . . . . . . . . . . . . . 9.1. Global invariance. . . . . . . . . . . . . . . . . . . . . . 9.2. Local invariance. . . . . . . . . . . . . . . . . . . . . . 9.3. Volterra-Lotka predator-prey model. . . . . . . . . . . 9.4. Pendulum. . . . . . . . . . . . . . . . . . . . . . . . . . 9.5. Emphasizing the proof. . . . . . . . . . . . . . . . . . . 9.6. Volterra-Lotka predator-prey model. . . . . . . . . . . 9.7. Non decomposable quadrics. . . . . . . . . . . . . . . . 10.1. Center manifold 2D. . . . . . . . . . . . . . . . . . . . 10.2. Center manifold 3D. . . . . . . . . . . . . . . . . . . . 10.3. Center manifold 2D depending upon a parameter. . . 10.4. Center manifold 3D depending upon a parameter. . . 10.5. Linking both transformations. . . . . . . . . . . . . . 11.1. Van der Pol model. . . . . . . . . . . . . . . . . . . . 11.2. Chua’s system. . . . . . . . . . . . . . . . . . . . . . . 12.1. Spherical pendulum. . . . . . . . . . . . . . . . . . . . 12.2. Van der Pol system. . . . . . . . . . . . . . . . . . . . 12.3. First integral of Volterra-Lotka system. . . . . . . . .

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71 71 77 80 82 87 88 91 93 94 97 99 101 104 106 107 112 117 119 136 141 144 146 147 151 152 153 154 156 162 166 169 174 180 193 198 204 204 208

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List of Examples

12.4. 12.5. 12.6. 13.1. 13.2. 13.3. 13.4.

First integral of quadratic system. Homogeneous dyanmical system. . Kapteyn-Bautin system. . . . . . . Van der Pol model. . . . . . . . . Lorenz model. . . . . . . . . . . . Van der Pol model. . . . . . . . . Flow curvature manifold parity. .

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xxvii

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210 212 213 216 217 220 221

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PART 1

Dynamical Systems

The true method of foreseeing the future of mathematics is to study its history and its actual state. — H. Poincaré —

1

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2

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Chapter 1

Differential Equations

“The scientist does not study nature because it is useful; he studies it because he delights in it, and he delights in it because it is beautiful. If nature were not beautiful, it would not be worth knowing, and if nature were not worth knowing, life would not be worth living.” — H. Poincaré —

The discovery of infinitesimal calculus at the end of the seventeenth century enabled to study natural phenomena which could be thus modeled through the agency of what were virtually differential equations. The term œquatio differentiale or differential equation was first used by Leibniz (1684) to denote a relationship between the differentials dx and dy of two variables x and y. Newton’s second law provided one of the most fruitful sources of differential equations, i.e. equations involving a function and its derivatives.

1.1

Galileo’s pendulum

According to Vincenzo Viviani (1622-1703), Galileo’s last disciple and first biographer, Galileo (1564-1642) had already empirically1 observed the isochronism of pendulums in 1581 when he was a 17-years-old student in Pisa. While he was attending a Mass in the Duomo of Pisa he noticed that a bronze chandelier or incense burner was swaying in the breeze, sometimes barely moving and other times swinging in a wide arc. He timed the swings with his pulse. To his surprise, it took the same number of pulse beats for the chandelier to complete one swing no matter how far it moved. The wider the swing, the faster the motion was, but always in the same amount 1 For

controversies about Galileo’s experiments see A. Koyré, T. B. Settle, S. Drake, I. B. Cohen.

3

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of time provided that amplitude of motion kept small.

Fig. 1.1

Galileo’s pendulum

Newton’s second law of motion enables a Mathematical description of this phenomenon and leads, in the small-amplitude approximation, to the d2 θ g following equation: 2 + θ = 0 which is a 2nd order ordinary differential dt L equation of the 1st degree. Ordinary differential equations (O.D.E.) express a relation between the derivatives of a dependent variable with respect to a single independent variable while partial differential equations (P.D.E.) express a relation between the derivatives of a dependent variable with respect to two or more independent variables. The order of a differential equation is the highest order of differentiation appearing in the equation and the degree of a differential equation written as a polynomial of all the derivatives is the power to which the highest derivative appearing in the equation is raised. Example 1.1. Order and degree of differential equations Fourier’s law of heat diffusion: ferential equation of 1st degree.

∂ 2T ∂T = κ 2 is a 2nd order partial dif∂t ∂x

Blasius equation arises which in the theory of fluid boundary layers: 2 dy d3 y d2 y +ay 2 = β − 1 is a 3nd order ordinary differential equation dx3 dx dx of 1st degree.

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1.2

DGeometry

5

D’Alembert transformation

In the middle of the seventieth century D’Alembert transformation enabled to transform any single nth order differential equation into a system of n simultaneous first-order equations, and conversely. Let’s consider the nth order differential equation: dχ d2 χ dn−1 χ dn χ = F (χ, , ..., , t) , dtn dt dt2 dtn−1 By posing: χ(t) = x1 (t), χ (t) = x2 (t), . . . , χ(n−1) (t) = xn (t) this equation is equivalent to a system of n simultaneous equations of the first order:          

dx1 = x2 dt ···

dxn−1  = xn     dt     dxn = F (x1 , x2 , ..., xn , t) dt

(1.1)

Proof. Cf. D’Alembert (1748) ; Coddington and Levinson (1955, p. 21) ; Ince (1926, p. 14) ; Petrovski (1966, p. 89) ; Arnold (1963, p. 100). Since t only appears as an intrinsic variable, it can be eliminated, and system (1.1) may be written in the so-called symmetric form (Poincaré (1886, p. 168) ; Nemytskii and Stepanov (1960, p. 35) ; Petrovski (1966, p. 91) ; Davis (1962, p. 17)): dx2 dxn dx1 = = ... = f1 f2 fn

(1.2)

Remark. The transformation of a nth order differential equation into a system of n simultaneous equations of the first order is not unique. Notation. In the following the dot (·) will represent the time derivative.

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6

Example 1.2. System of differential equations d2 x + kx = 0 which is a 2nd order dt2 O.D.E. of the 1st degree may be transformed into the following system of two simultaneous equations of the first order while posing ω 2 = k/m: Hooke’s law of elastic restoration: m

2

m

d x + kx = 0 dt2

⇔

 1  dx dt = x2 2 2  dx dt = −ω x1

⇔

 2 1  dx dt = −ω x2 2  dx dt = x1

1.3

From differential equations to dynamical systems

According to D’Alembert’s transformation and following the definition proposed by G. D. Birkhoff (1912, p. 306) of his memoir originally presented in 1909 at a meeting of the American Mathematical Society and entitled: Quelques Théorèmes sur le mouvement des systèmes dynamiques. “A dynamical system in a very large meaning may be considered as being defined by any system of differential equations of the first order: dx2 dxn dx1 = = ··· = = dt X1 X2 Xn where X1 , . . . , Xn are given functions, real and uniform depending on x1 , . . . , xn , analytical with respect of these variables, and where t is the independent variable. Variables x1 , . . . , xn are the coordinates of the motion and t indicates the time.”

This is exactly the same definition as previously proposed by Henri Poincaré (1886, p. 168) in one of his famous memoirs entitled: Sur les courbes définies par une équation différentielle. “. . . any differential equation can be written as: dx1 = X1 , dt

dx2 = X2 , dt

...,

dxn = Xn dt

where Xi are real polynomials. If t is considered as the time, these equations will define the motion of a variable point in a space of dimension n.”

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Chapter 2

Dynamical Systems

“One would have to have completely forgotten the history of science so as to not remember that the desire to know nature has had the most constant and the happiest influence on the development of mathematics.” — H. Poincaré —

The notion of dynamical system is the mathematical description of the dynamics of a given physical, mechanical, electronic, biological, ecological, economical system from the point of view of a deterministic process which is expressed in terms of state variables, making it possible to define the instantaneous state of the system, and equations of evolution of these variables between an initial and final instant. According to whether these instants are separated by a finite or infinitesimal time, equations of evolution are mapping iterations (Collet and Eckman, 1980) or differential equations (Hirsch and Smale, 1974). In the former case, the devoted terminology is discrete-time dynamical system, in the latter, continuous-time dynamical system or dynamical system. This work deals more particularly with autonomous dynamical systems, i.e. systems of differential equations in which time does not appear explicitly.

“The key to the geometric theory of dynamical systems created by Henri Poincaré, is the phase portrait of a dynamical system. The first step in drawing this portrait is the creation of a geometric model for the set of all possible states of the system. This is called the state space 1 .”

1 Abraham

et al., 1982, Dynamics: the Geometry of behavior (Vol. 1) p. 11 7

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8

2.1

State space – phase space

= {xi ; i = 1, 2, ..., n} the state variables set of a dynamiLet’s denote: X cal system which entirely defines its instantaneous state. The state variables set is used as a canonical coordinates system in a n-dimensional space called state space or phase space following thus a tradition from classical mechanics. The number of components of the state variables set, i.e. n, represents the dimension of the dynamical system and is also called the degree of freedom. The concept of phase plane which derives from Statistical Mechanics has been originally defined by Henri Poincaré (1886, p. 168) as follows: “Considering x and y as the coordinates of a variable point, and t as the time, one seeks the motion of a point to which one gives the velocity as a function of the coordinates.”

2.2

Definition

Let’s consider a system of differential equations defined in a compact E = [x1 , x2 , ..., xn ]t ∈ E ⊂ Rn : included in Rn with X dX X) = ( dt

(2.1)

X) = [f1 (X), f2 (X), ..., fn (X)] t ∈ E ⊂ Rn defines in E a velocity where ( vector field whose components fi , supposed to be C ∞ continuous functions in E with values in R, are checking the assumptions of the Cauchy-Lipshitz theorem (Cf. supra). A solution of this system is the parametrized tra jectory curve X(t) whose values are defining the states of the dynamical system (2.1). If each component of the velocity vector field does not explicitly depend on time the systems are said to be autonomous.

2.3

Existence and uniqueness

Theorem 2.1 (Cauchy). Consider the initial value problem: dX X), X(t 0) = X 0 with X 0 ∈ E ⊂ Rn . = ( dt : Rn → Rn is a C 1 function in the vicinity of X 0 then there exists If

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9

a unique solution of this initial value problem. So, there exists α > 0 : (t0 − α, t0 + α) → Rn of this differential system and a unique solution X 0. (t0 ) = X satisfying the initial condition X Proof. Cf. Cauchy (1835) ; Hirsch et al. (2004, p. 144) ; Coddington and Levinson (1955, pp. 11-13) ; Ince (1926, p. 76 and next) ; Davis (1962, p. 79) ; Petrovski (1966, p. 89) ; Arnold (1974, p. 34). Remark. Many different types of existence theorem are generally established and usually referred in the previous quoted literature as: Calculus of Limits, Method of Successive Approximations, Cauchy-Lipschitz method. The name “the calculus of limits” has been given to an existence theorem originally contributed by A. L. Cauchy (1789-1857), which marked the first proof of existence and uniqueness of a solution of (2.1), subject to initial X) 0 and where ( is of a rather general form2 . (t0 ) = X condition X

2.4

Flow, fixed points and null-clines

dX X), = ( dt ∈ E ⊂ Rn and X( X 0 , t), X 0 ∈ D, a trajectory curve of (2.1) with X X) generates a flow the initial condition X (t0 ) = X0 , the vector field ( n Φt : D → R defined by Φt (X0 ) = X(X0 , t) and satisfying the properties: Definition 2.1. Let’s consider the dynamical system (2.1)

0 ) is a C r function, (i) Φt (X 0) = X 0, (ii) Φ0 (X 0 ) = Φt (Φs (X 0 )). (iii) Φt+s (X MF 01

Example 2.1. Free fall According to Galiléo Galilé¨ı (1564-1642), acceleration of falling bodies in free fall is constant in the neighborhood of the ground. If z is the height from which the body falls without initial velocity and with the nominal 2 When in 1835 the French Academy of Sciences created the Comptes Rendus des s´ eances hebdomadaires which became a century later, the Comptes rendus de l’Académie des sciences (Proceedings of the French Academy of Sciences), Cauchy overwhelmed it every week with very long memoirs. Then, the Academy decided to institute a rule, still in current, imposing that the number of pages should not exceed four printed pages. This anecdote is related in Bell (1937, p. 132).

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acceleration g due to gravity at the Earth’s surface at sea level3 , Newton’s d2 z = −g. law of motion reads: dt2 D’Alembert tranformation leads to the following dynamical system in the phase plane: dX dt

dx

dt dy dt

=

f (x, y)

=

g (x, y)

y

−g

With the initial conditions x (0) = 0 and x˙ (0) = 0 equivalent to x (0) = 1 0 and y (0) = 0, the flow of this system reads: (x (t) , y (t)) = − gt2 , −gt 2 or y 2 +2gx = C where C = y 2 (0)+2gx (0). The flow of this system has been plotted in Fig. 2.1 for various initial conditions: (x (0) , y (0)) = (±0.8k, 4) where k = 0, 1, 2, 3, 4. Y 4

2

0

X

2

4 4

2

Fig. 2.1

0

2

4

Free fall

Definition 2.2. Fixed points, also called equilibria or singular points, de ∗ are points of the phase space defined by vanishing the vector noted X X i.e. ( ∗ ) = 0. So, in such points the tangent to the trajectory field (X), curve can not be defined. 3g

= 9.80665 m.s−2 strictly means the local acceleration due to gravity which varies depending on one’s position (latitude) on Earth.

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11

dX X), = ( dt X) = [x1 , ..., xn ]t ∈ E ⊂ Rn and ( = [f1 (X), ..., fn (X)] t ∈ E ⊂ Rn , X = 0 with the xj -null-clines is the set of points determined by setting fj (X) j = 1, ..., n, and the intersection of which provide the fixed points.

Definition 2.3. For any dynamical system (2.1) defined by

Remark. In the beginning of the twentieth century many techniques have been developed in order to plot the solution of a differential equation, i.e. a dynamical system, in the phase plane. In his famous article, On Relaxation Oscillations4 , Balthazar Van der Pol, inaugurated the method of “isoclynes”. “Let us draw in a z, v plane (phase plane) a series of “isoclynes”, i.e. dz is equal to a certain quantity. curves connecting all points for which dv dz = C1 where C1 is a An example of such isoclynes is denoted by dv constant (. . . ). Several of these isoclynes may be drawn in the z, v plane, and we can indicate by means of short lines the direction the integral curve must have when it crosses an isoclyne. From a diagram in which the inclinations of the integral curves the z, v plane.”

Thus, long beforehand computers development, Van der Pol method of “isoclynes” provided a graphical method of integration of differential equations, i.e. of dynamical systems. MF 02

Example 2.2. Volterra-Lotka predator-prey model 5 In the beginning of the twentieth century the Italian Mathematician Vito Volterra (1860-1940) was questioned by his son-in-law the Zoologist Umberto D’Ancona (1896-1964) on the problem of a great increase of certain species of fish (selacians) during the first world war in three harbor of the Adriatic (Trieste, Fiume and Venice). D’Ancona which was dealing with fishing statistics in these harbors had noticed that during the period 1915-1921 the number of selacians fished had raised up in important proportion. He made the assumption that fishing was perturbing the natural equilibrium between species and he asked his father-in-law to find a mathematical demonstration of his hypothesis corresponding to this observed phenomenon. The result was the so-called Volterra-Lotka predator-prey model (Cf. Fig. 2.2). Just few years before the American statistician Alfred Lotka had proposed a model describing an “ideal” oscillatory chemical 4 Van

der Pol (1926, p. 982) (1926) ; Lotka (1925)

5 Volterra

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12

reaction which lead to a predator-prey model. Nevertheless6 , the following model constitutes a very small part of the great contribution of Vito Volterra in this domain since he had elaborated in his book7 what is actually considered as the foundations of Mathematical Ecology.

6

5

4

3

2

1

0 0

Fig. 2.2

1

2

3

4

5

6

Volterra-Lotka predator-prey model

The null-clines of the Volterra-Lotka predator-prey model are given by:

f (x, y) = 0 g (x, y) = 0

⇔

x (a − by) = 0 y (dx − c) = 0

Intersections of four straight lines: x = 0, x = c/d (vertical), y = 0, y = a/b (horizontal) provide two fixed points: O (0, 0), I(c/d, a/b). Remark. Null-clines are playing a very important role in the study of certain kinds of dynamical systems, namely singularly perturbed systems also called slow-fast autonomous dynamical systems (Cf. supra). Moreover, = α, where α is a real parameter isoclines determined by setting fj (X) arbitrary chosen, may be also used. 6 For

details about the “ quarrel of priority ”, see Israel (1996, p. 67) (1926, 1928, 1931)

7 Volterra

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13

Stability theorems

2.5.1

Linearized system

dX X) with X ∈ E ⊂ Rn . = ( dt X) in the vicinity of the fixed The Taylor expansion of the vector field ( ∗ point X reads:

Let’s consider the dynamical system (2.1):

X) X X = ( ∗ ) + D ( −X ∗ ) + O (X −X ∗ )2 ∗ )(X ( X X) is the functional = ( ∂fi (X) ) = J(X) Let’s notice that matrix DX ( ∂xj Jacobian matrix associated with the dynamical system and O(εk ) represents a real polynomial in ε of degree k in Landau’s notation. Hartman-Grobman linearization theorem dX X), X ∈ E ⊂ Rn has = ( Suppose that the dynamical system (2.1): dt ∗ so that none eigenvalues of the functional Jacobian matrix a fixed point X ∗ is hyperbolic. Then the nonlinear flow ∗ ) has real part null, i.e. X J(X is locally topologically conjugate to the flow of the linearized system in the ∗ , i.e. there is a homeomorphism8 h : Rn → Rn mapping orbits vicinity of X of the nonlinear flow to those of the linear flow, preserving time direction. 2.5.2

Proof. 2.5.3

dX X) = J(X ∗ )(X −X ∗) = ( dt Cf. Hartman (1964).

(2.2)

Liapounoff stability theorem

dX X), X ∈ E ⊂ Rn = ( dt X ∗ such that: ( ∗ ) = 0. Let’s denote J(X ∗ ) its with a fixed point X ∗ is ∗ . Then X functional Jacobian matrix evaluated at the fixed point X ∗ stable if all eigenvalues λ1 , λ2 , ..., λn of J(X ) have negative real part, i.e. iff Re(λi ) < 0. Let’s consider the dynamical system (2.1):

Proof. 8 See

Cf. Liapounoff (1899).

Appendix for definitions of homeomorphism and diffeomorphism.

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14

2.6

Phase portraits of dynamical systems

According to Liapounoff stability theorem the sign of the real part of the eigenvalues associated with the functional Jacobian matrix evaluated at the fixed point enables to define its stability. 2.6.1

Two-dimensional systems

dX X), = ( Lets’ consider a two-dimensional dynamical system (2.1): dt

t X) = [x, y]t ∈ E ⊂ R2 , ( = f (X), g(X) ∗ ) its X ∈ E ⊂ R2 and J(X ∗: functional Jacobian matrix evaluated at the fixed point X  ∂f ∗) =  J(X

∂f ∂x ∂y

∂g ∂g ∂x ∂y

 

(2.3)

∗ X=X

Eigenvalues λi are roots of the Cayley-Hamilton characteristic equation: λ2 − T r (J) λ + Det (J) = 0

(2.4)

Sum and product of the eigenvalues are given by the trace and the determinant of the functional Jacobian matrix, respectively setting: p = T r (J) =

∂f ∂g + = λ1 + λ2 ∂x ∂y X=

X

∗

and q = Det (J) =

∂f ∂g ∂f ∂g − = λ1 λ2 ∂x ∂y ∂y ∂x X=

X

∗

nature of the eigenvalues is given by the discriminant: 2

∆ = p2 − 4q = T r (J) − 4Det (J)

(2.5)

Then, three cases are to be considered for which a classification has been introduced by Henri Poincaré in his Ph-D thesis (1879) and reproduced in his memoirs (Poincaré (1881, p. 387 and next)).

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15

• If ∆ > 0 both eigenvalues are real and distinct. • If ∆ = 0 both eigenvalues are real and repeated. • If ∆ < 0 both roots are complex conjugated. This leads to five sub-cases9 (i) (ii) (iii) (iv) (v)

Both Both Both Both Both

roots roots roots roots roots

real and positive. real but differing in sign. real and negative. conjugated complex numbers. pure imaginary.

In Fig. 2.3 the next results have been plotted in the (p, q) plane. Thus, starting from the lower right side of Fig. 2.1 (case (i)) and following the clockwise direction all others cases may be easily identified. While case (v), i.e. the center is in the middle of the figure. (i) ∆ ≥ 0, q > 0 and p > 0. ∗ is an If both eigenvalues are real and positive then the fixed point X asymptotically unstable node. (ii) ∆ > 0, q < 0. ∗ is If both eigenvalues are real of opposite sign then the fixed point X a saddle. (iii) ∆ ≥ 0, q > 0 and p < 0. ∗ is an If both eigenvalues are real and negative then the fixed point X asymptotically stable node. (iv) ∆ < 0. If both eigenvalues are complex conjugated and p < 0 (resp. p > 0) ∗ is an asymptotically stable (resp. unstable) then the fixed point X focus. (v) ∆ < 0, p = 0. ∗ is a If both eigenvalues are pure imaginary then the fixed point X center, stable but not asymptotically stable. 9 In Appendix it will be shown that there is a matrix called Jordan form corresponding to each of these cases.

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q

p

Fig. 2.3

Phase plane stability diagram

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MF 03

17

Example 2.3. Inverted pendulum In the small-amplitude approximation, Newton’s second law of motion d2 θ g − θ = 0. The corresponding dynamleads to the following equation: dt2 L ical system may be written in the phase plane: dX dt

dx

dt dy dt

=

f (x, y) g (x, y)

=

y x

∗ (0, 0) reads: The Jacobian matrix evaluated at the fixed point X ∗) = J(X

0 1 1 0 X=

∗

X

Thus, p = T r (J) = 0, q = Det (J) = −1 < 0 and so, ∆ = 4 > 0. Both eigenvalues are real of opposite sign (λ1,2 = ±1) then the fixed point ∗ (0, 0) is a saddle (case (iii)). X Y 4

2

0

X

2

4 4

2

Fig. 2.4

0

2

4

Inverted pendulum

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18

2.6.2

Three-dimensional systems

dX X) = ( Let’s consider a three-dimensional dynamical system (2.1):

t dt X) = [x, y, z]t ∈ E ⊂ R3 , ( = f (X), g(X), h(X) with X ∈ E ⊂ R3 and ∗: ∗ ) its functional Jacobian matrix evaluated at the fixed point X J(X  ∂f  ∗) =  J(X 

∂x ∂g ∂x ∂h ∂x

∂f ∂y ∂g ∂y ∂h ∂y

∂f ∂z ∂g ∂z ∂h ∂z

   

(2.6)

X

∗ X=

Eigenvalues λi are roots of the Cayley-Hamilton characteristic: λ3 − T r (J) λ + M (J) λ − Det (J) = 0

(2.7)

Sum and product of the eigenvalues are given by the trace and the determinant: p = T r (J) = λ1 + λ2 + λ3

;

q = Det (J) = λ1 λ2 λ3

Minors of the functional Jacobian matrix reads: r = M (J) = λ1 λ2 + λ2 λ3 + λ1 λ3 where M (J) = J11 + J22 + J33 and Jii is the minor obtained by deleting the ith row and the ith column of the functional Jacobian matrix. Nature of eigenvalues is given by the discriminant of Girolamo Cardano (1501-1576): R = 4P 3 + 27Q2 where P = r −

p2 3

(2.8) 3

and Q = − 2p 27 +

pr 3

−q

Then, three cases are to be considered for which a classification has been introduced by Henri Poincaré in his memoirs entitled Sur les courbes définies par une equation différentielle (Poincaré (1886, p. 166 and next)).

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19

• If R < 0 both eigenvalues are real and distinct. • If R = 0 two eigenvalues are real and repeated, one is simple real. • If R > 0 two roots are complex conjugated and one is real. This leads to five sub-cases10 (i) Both roots real and positive. (ii) Both roots real and negative. (iii) Both roots real but differing in sign. (iv) Two roots conjugated complex numbers. (v) Two roots pure imaginary. The newt results have been plotted in Fig. 2.5. (i) R ≤ 0, q > 0 and p > 0 ∗ is an If both eigenvalues are real and positive then the fixed point X unstable node. (ii) R ≤ 0, q > 0 and p < 0 ∗ is a If both eigenvalues are real and negative then the fixed point X stable node. (iii) R ≤ 0 ∗ is If both eigenvalues are real of opposite sign then the fixed point X a saddle. (iv) R > 0 If two eigenvalues are complex conjugated and one is real and positive ∗ is a focus (resp. saddle focus). (resp. negative) then the fixed point X (v) R > 0 ∗ is a If both eigenvalues are pure imaginary then the fixed point X center.

10 In appendix it will be shown that there is a matrix called Jordan form corresponding to each of these cases.

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Fig. 2.5

Stability diagram

20

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21

Example 2.4. Saddle-focus Let’s consider the following dynamical system:   dX  dt 

dx dt dy dt dz dt





f (x, y, z)





−x + y



      g (x, y, z)  =  −x − y  =      −2z h (x, y, z)

Sum and product of the eigenvalues given by the trace, determinant and minors of the functional Jacobian matrix read: p = T r (J) = −4 ; q = Det (J) = −4 ; r = M (J) = +6. The discriminant of the CayleyHamilton characteristic equation provides the nature of eigenvalues, i.e. 2 2p3 pr 20 p2 R = 4P 3 + 27Q2 = 16 > 0 with P = r − = ;Q=− + −q = . 3 3 27 3 27 So, there are two complex conjugated eigenvalues and one real and negative ∗ (0, 0, 0) is a saddle focus. (λ1,2 = −1 ± i, λ3 = −2). Thus, the fixed point X According to Poincaré (1886, p. 166) and to Arrowsmith and Place (1982, p. 65) all trajectories are lying on surfaces of equation: z = x2 + y 2 while their projection onto a plane of constant z is a stable focus.

Fig. 2.6

Saddle-focus

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2.7 2.7.1


DGeometry


Various types of dynamical systems Linear and nonlinear dynamical systems

Definition 2.4. A dynamical system defined by (2.1) is linear if all com X) are linear functions of xi . Otherwise ponents fi of the vector field ( the dynamical system is said to be nonlinear. Example 2.5. Linear and nonlinear systems Inverted pendulum is an example of linear dynamical system while Volterra-Lotka predator-prey model is nonlinear because of the terms of predation: xy. Remark. There exists also other kinds of dynamical systems such piecewise linear dynamical systems e.g., Chua’s model may be viewed as the paradigm of piecewise linear systems (Cf. supra). 2.7.2

Homogeneous dynamical systems

X) is an homogeneous polynomial If each component fi of the vector field ( in xi , then the dynamical system is said to be homogeneous and denoted X) is an homogeneous polynomial (H.D.S.). If each component fi of ( ∗ : Rn → Rn can be of same degree p (p ∈ N ) in xi , then the mapping 1 written in the matrix form (X) = p J X where J is the functional Jacobian matrix. Moreover if p = 1, then the system becomes linear and is called Jacobi’s system. 2.7.3

Polynomial dynamical systems

X) is a polynomial of degree If each component fi of the vector field ( p in xi , then the dynamical system is said to be polynomial and denoted X) is a polynomial of same degree (P.D.S.). If each component fi of ( ∗ p (p ∈ N ) in xi , then the mapping : Rn → Rn can be written in the +X 0 where Jp is the functional Jacobian matrix X) = 1 Jp X matrix form ( p 0) = [c1 , c2 , ..., cn ]t ∈ E ⊂ 0 = ( associated to the term of degree p and X Rn with ci constant. Moreover, if all ci are vanishing, the system becomes homogeneous again.

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23

Example 2.6. Quadratic systems Let’s consider the following quadratic system: dX dt

dx

dt dy dt

= =

f (x, y) g (x, y)

X + 1 J2 (X) X +X 0 = J1 (X) 2

a00 + a10 x + a01 y + a20 x2 + a11 xy + a02 y 2

b00 + b10 x + b01 y + b20 x2 + b11 xy + b02 y 2

where aij and bij with i, j = 0, 1, 2 are real parameters. Such system is polynomial since it consists of polynomial function of degree two in xi . If both a00 = a10 = a01 = 0 and b00 = b10 = b01 = 0 the quadratic system becomes homogeneous since the transformation x → kx, y → ky leads to f (x, y) → k 2 f (x, y) and g (x, y) → k 2 g (x, y) for any parameter k called the weight. Then, the system may be written: dX dt

dx

dt dy dt

= 1 = 2

f (x, y) g (x, y)

=

1 J(X)X 2

2a20 x + a11 y a11 x + 2a02 y 2b20 x + b11 y b11 x + 2b02 y

x y

If one of the parameters {a00 , a10 , a01 , b00 , b10 , b01 } is not identically zero the system is still polynomial but not homogeneous anymore and so can not be written as previously. 2.7.4

Singularly perturbed systems

A dynamical system having a small multiplicative parameter ε ∈ I = [0, 1] ⊂ Rn in factor in one or several component fi of its vector field X), and a singular approximation (Cf. supra) is said to be singularly ( perturbed. Example 2.7. Slow-fast dynamical system Van der Pol model which may be considered as the paradigm of Relaxation Oscillations is a polynomial slow-fast dynamical system (Cf. supra).

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24

2.7.5

Slow-Fast dynamical systems

A dynamical system having no small multiplicative parameter in factor in the component of its vector field, and no singular approximation (Cf. supra) is said to be slow-fast if its functional jacobian matrix has at least a “fast” eigenvalue, i.e. with the largest absolute value of the real part. Remark. A singularly perturbed system is slow-fast. A slow-fast system is not necessary singularly perturbed (Cf. supra). 2.8

Two-dimensional dynamical systems

In the first and second part of his memoirs entitled, Sur les courbes définies par une équation différentielle, Poincaré (1881, 1882) introduced many useful concepts recalled in this section. Let’s consider a two-dimensional dynamical system: dX X) = ( (2.9) dt

t X) = [x, y]t ∈ E ⊂ R2 , ( = f (X), g(X) X ∈ E ⊂ R2 where f and g ∞ are supposed to be C continuous functions in E with values in R, checking the assumptions of the Cauchy-Lipschitz theorem (Cf. infra). 2.8.1

Poincar´ e index

Let’s consider a closed, differentiable curves φ (x, y) = C not passing through any fixed point of the dynamical system (2.9). If φ (x, y) = C ∗ , then k the is built such in order to encircle a single isolated fixed point, X 11 index of this closed curve is defined by: 1 k= 2π

g (x, y) f dg − gdf 1 dArctan = f (x, y) 2π f 2 + g2

φ

11 k

is independent of the form of the closed curve.

φ

DGeometry

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Theorem 2.2 (Poincar´ e). Index of a closed curve (i) Index of a closed curve not containing any fixed point is zero. (ii) Index of a closed curve containing a focus, a center or a node is +1. (iii) Index of a closed curve containing a hyperbolic saddle is −1. Proof.

Cf. Poincaré (1881, p. 401 and next).

Corollary 2.1. The total number of nodes (N) and foci (F) is equal to the total number of saddles (S) plus two. N +F =S+2 Proof.

(2.10)


Application to linearized system 12 Let’s consider the two-dimensional dynamical system linearized13 in the ∗ (0, 0) reading: vicinity of its fixed point X dX dt

dx dt dy dt

=

f (x, y) g (x, y)

=

λ1 0 0 λ2

x λ1 y = λ2 x y

where λi are the real eigenvalues of the functional Jacobian matrix. Thus ∗ (0, 0) is given by: it may be checked that the index k of the fixed point X

k=

1 2π

d arctan φ

λ1 y λ1 λ2 (xdy − ydx) 1 = 2 2 λ2 x 2π (λ1 y) + (λ2 x) φ

By selecting the closed curve as an ellipse: |λ2 | x = cos (t), |λ1 | y = sin (t) we have

k=

λ1 λ2 = |λ1 λ2 |

+1, λ1 λ2 > 0 −1, λ1 λ2 < 0

12 This 13 See

presentation is due to Hochsdadt (1963, p. 273). Appendix for Jordan form.

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26

Example 2.8. Van der Pol model (1926) Let’s consider the model of Van der Pol (1926) for which µ = 0:

V

x˙ f (x, y) y = = y˙ g (x, y) −x + µ 1 − x2 y

∗ (0, 0) The functional Jacobian matrix evaluated at the fixed point X 0 1 reads: J = . Thus, according to Sec. (2.6.1) the product λ1 λ2 = −1 µ ∗ (0, 0) is k = +1. Hence Det (J) = +1 and so the index of the fixed point X this fixed point is either a focus, or a node. It can be easily checked that its nature depends on the parameter value µ. Moreover, since it has been ∗ (0, 0) can not be a center. supposed that µ = 0, the fixed point X Poincar´ e contact theory

2.8.2

In chapter IV of his first memoir14 Poincaré defined a topographical system 15 of nonintersecting, closed, differentiable curves φ (x, y) = C. Contact curves consists of points at which the curves of the topographical system are tangent to the trajectory curves of the dynamical system: dX X) = ( dt

t = [x, y]t ∈ E ⊂ R2 and ( X) = f (X), g(X) with X ∈ E ⊂ R2 . Thus contact curves F (x, y) = C satisfy the equation16 :

f (x, y)

∂φ ∂φ + g (x, y) =0 ∂x ∂y

⇔

=0 LV φ(X)

(2.11)

and will enable to establish the existence of limit cycles. 14 Journ. 15 Such

Math. Pures Appl., 1881, (3) 7, 375-422, p. 409 system is called topographical system by analogy with the level curves system of

a map. 16 See Appendix for Lie derivative definition.

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Poincar´ e limit cycle

2.8.3

The concept of limit cycle was introduced by Henri Poincaré in his memoir Sur les courbes définies par une équation différentielle (Journ. Math. Pures Appl., 1882, (3) 8, 251-296, p. 263). Existence of limit cycles was established while using the theory of limits rings. Theorem 2.3. Let’s consider a topographical family of concentric circles: φ (x, y) = x2 + y 2 = r2 ≥ 0 for If there exist two radii r0 , r1 such that r0 < r1 and LV φ(X) 2 2 2 2 2 ≤ 0 for x + y = r and if there is no fixed x + y = r0 and LV φ(X) 1 points in the ring-shaped region: Γ : r02 ≤ x2 + y 2 ≤ r12 , then Γ contains a is negative on the inner circle and is positive stable limit cycle. If LV φ(X) on the outer circle, then Γ contains a unstable limit cycle.17 2

Proof.


Corollary 2.2. Inside and outside any limit cycle there is always at least a focus or a node. Proof.


Remark. It can be deduced from Corollary 2.2 that the unique fixed point ∗ (0, 0) of the Van der Pol model (Ex. 2.8) is either a focus or a node. X MF 05

Example 2.9. Historically first limit cycle (Poincaré, 1882, p. 278) The following dynamical system is the first example originally used by H. Poincaré in order to highlight the concept of limit cycle: V

2 x˙ f (x, y) x x + y 2 − 1 − y x2 + y 2 + 1 = = y x2 + y 2 − 1 + x x2 + y 2 + 1 y˙ g (x, y)

While considering the topographical family of concentric circles: φ (x, y) = x2 + y 2 = r2 with radii r0 = 1/2, r1 = 3/2 (for example) ≤ 0 for x2 + y 2 = r02 and L φ(X) ≥ 0 it’s easy to check that LV φ(X) V 2 2 2 2 2 2 2 for x + y = r1 in the ring-shaped region: Γ: r0 ≤ x + y ≤ r1 . Then, according to Th. 2.3, Γ contains an unstable (repulsive) limit cycle whose equation is that of the unit circle: x2 + y 2 = 1. 17 This

presentation is due to Nemytskii and Stepanov (1960, p. 138).

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2

1

0

1

2 2

1

Fig. 2.7

0

1

2

Poincaré limit cycle

Moreover, it can be deduced from Corollary 2.2 that the fixed point ∗ X (0, 0) is either a focus or a node. It is important to notice that this first limit cycle which was built in an ad hoc manner by Poincaré in order to emphasize its concept is algebraic. But as pointed out in Poincaré (1882, p. 283): “When limit cycles are not algebraic, a complete discussion is obviously not possible; since one will never find in finite terms the equation of limit cycles.”

This premonitory thinking has been verified in the case of Van der Pol model by Yuri Mitropolskii18 and later by Kenzi Odani19 . The former has shown that the Van der Pol model is not integrable (Cf. supra) while the latter has established that the limit cycle of this model is not algebraic. 18 Mitropolski, Yu. A. (1965) Problems of the asymptotic theory of nonstationary vibrations, D. Davey Press, New York (Translated from Russian) 19 Odani, K. (1995) “The limit cycle of the van der Pol equation is not algebraic,” J. Diff. Eqs. 115, 146-152

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29

Poincar´ e-Bendixson Theorem

Necessary and sufficient conditions for the existence of limit cycles have been also given by Ivar Bendixson in his memoir Sur les courbes définies par des équation différentielles (Acta Mathematica, 1899, 24, 1-88, p. 11). Theorem 2.4. Poincaré-Bendixson Theorem (t) solution to the two-dimensional Suppose20 that the trajectory curve X dynamical system (2.9) is contained in a bounded region D ⊂ R2 of the (t) must either: phase plane for t ≥ 0. Then, as t → +∞, X (i) tends to a fixed point; or (ii) spirals towards a limit cycle of dynamical system (2.9). Proof. Cf. Bendixson (1901, p. 11 and next) ; Coddington and Levinson (1955, p. 391 and next) ; Hirsch et al. (2004, p. 225 and next). Remark. This theorem is based on the fact that a closed curve in the plane (with no self-intersections, i.e. no double points) divides the plane into two parts. This is no longer true in higher dimension and there is no analogue of the theorem in dimension three or more. In the same memoir Bendixson (1901, p. 78) introduced a criterion making it possible to establish the non-existence of limit cycles for twodimensional dynamical systems. Theorem 2.5. Bendixson’s criterion Let D ⊂ R2 be a simply connected 21 region of the phase plane in which X) = [f (X), g(X)] t ∈ E ⊂ R2 has the property that the vector field ( ∂g ∂f · + =∇ ∂x ∂y is not identically zero and does not change its sign, then the dynamical system (2.9) has any closed trajectory which lies entirely in D. Proof.

20 This 21 See

Cf. Bendixson (1901, p. 78 and next)

presentation is due to Arrowsmith and Place (1982, p. 110). Appendix for definition of connected region.

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30

Remark. As it will be pointed out in Ex. 2.10, Bendixson’s criterion does not prevent from the existence of a closed trajectory crossing simply connected region in which the criterion holds that’s the reason why Th. 2.4 is frequently referred to as a negative criterion. MF 06

Example 2.10. Duffing oscillator 22 Let’s consider the Unforced Duffing Oscillator

V

x˙ f (x, y) y = = y˙ g (x, y) x − x3 − δy + x2 y

This model has three fixed points O (0, 0) is a saddle, while I± (±1, 0) = x2 − δ. · are center and ∇ Y

4

2

0

X

2

4

2

1

Fig. 2.8

0

1

2

Duffing oscillator

Thus, there are two straight lines dividing the plane into three domains D1 , D2 and D3 . According to the Bendixson’s criterion such system would not have any periodic orbit entirely contained in one of these domains. But, it could have a periodic orbit crossing these domains (Cf. Fig. 2.8). 22 See

Wiggins (1990, p. 26 and next).

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2.9

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31

High-dimensional dynamical systems

According to Ruelle and Takens (1971) starting from dimension three, trajectory curves integral of three-dimensional dynamical systems may exhibit chaotic behavior leading to the existence of attractors23 . The simplest attractor is a fixed point. Another kind of attractor is the limit cycle and although there is no analogue for the Poincaré-Bendixson’s theorem for proving their existence high-dimensional dynamical systems may have limit cycles. 2.9.1

Attractors

Definition 2.5. Let’s consider a system of differential equations defined in = [x1 , x2 , ..., xn ]t ∈ E ⊂ Rn : a compact E included in R with X dX X) = ( (2.12) dt X) = [f1 (X), f2 (X), ..., fn (X)] t ∈ E ⊂ Rn defines in E a where ( vector field whose components fi , supposed to be C ∞ continuous functions in E with values in R, are checking the assumptions of the Cauchy-Lipshitz theorem with the flow Φt . A set Λ is called an attractor if: • Λ is compact and invariant. ∈ Λ, Φt (X) ∈U • There is an open set U containing Λ such that for each X for all t ≥ 0 and ∩t≥0 Φt (U ) = Λ. 1, X 2 ∈ Λ and any open neighbour• (Transitivity) Given any points X hoods Uj about Xj in U , there is a trajectory curve that begins in U1 and later passes through U2 . Remark. The set ∪t 0 with only slight distortion. This result has been stated by Arnold (1978) in the following theorem. Theorem 2.8. K.A.M. theorem Let’s suppose that a Hamiltonian system is given by H = H0 +εH1 where ε is a small parameter. H0 is integrable but the presence of H1 makes H nonintegrable. The quasiperiodic cycles, i.e. the K.A.M. tori which exist for ε = 0, will also exist for 0 < ε 1 but will be deformed by the perturbation. The K.A.M. tori dissolve one by one as ε increases. Proof. next).

Cf. Arnold (1978) ; Guckenheimer and Holmes (1983, p. 212 and

If each torus contains trajectory curves with two independent frequencies whose ratio is rational a resonance occurs between the two motions. At these resonances are saddle points, i.e. hyperbolic points with homoclinic or heteroclinic trajectories and tangles leading to the destruction of torus and irregular pattern of trajectory curves.

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38

MF 09

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Example 2.14. Hénon-Heiles Hamiltonian (Hénon, 1964) Let’s consider the following Hamiltonian consisting of two simple harmonic oscillators coupled with a cubic term: q 2 + q22 p21 + p22 q2 + 1 + q12 q2 − 2 2 2 3 This system has two degrees of freedom so Hamilton’s equations (2.15) read: H=

q¨1 + q1 = −2q1 q2 q¨2 + q2 = −q12 + q22

Transformed into a four-dimensional dynamical system:       x˙1 x3 f1 (x1 , x2 , x3 , x4 )       x4 f2 (x1 , x2 , x3 , x4 ) =  x˙2  =  V x˙3  f3 (x1 , x2 , x3 , x4 )  −x1 − 2x1 x2  x˙4 −x2 − x21 + x22 f4 (x1 , x2 , x3 , x4 ) this Hamiltonian system is not integrable in this case (Cf. supra). As the energy levels increase, many tori have disappeared and the closed trajectory curves, representing quasiperiodic behavior, are replaced by irregular patterns.

Fig. 2.11 Hénon-Heiles Hamiltonian with initial conditions p01 = −0.1, p02 = −0.2, q10 = 0.25, q20 = −0.05.

DGeometry

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39

By choosing various initial conditions (p01 , p02 , q10 , q20 ) for the same Hamiltonian H = 1/12 enables to generate other K.A.M. tori in the transversal Poincaré section x = 0 of the Hénon-Heiles Hamiltonian system.

Fig. 2.12

Transversal Poincaré section (p2 , q2 ) of H´ enon-Heiles Hamiltonian

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Chapter 3

Invariant Sets

“To invent is to discern, to choose.” — H. Poincaré —

Invariant sets play a very important role in dynamical systems theory since they are involved at different levels such as homoclinic bifurcations, Lasalle’s Invariance Principle, Attractors structure, integrability.... In this chapter many definitions, which will be useful in part two of this work, are recalled and illustrated. 3.1

Manifold

The concept of manifold which occurs in various contexts such as stable manifold theorem, centre manifold theory, invariant manifolds is defined as follows. 3.1.1

Definition

Let’s consider the manifold M ⊂ Rn as a set of points in Rn that satisfy a system of m scalar equations: =0 φ(X)

(3.1)

= [x1 , x2 , . . . , xn ]t ∈ E ⊂ Rn . The where φ : Rn → Rm for m ≤ n and X manifold M is smooth (differentiable) if φ is smooth and the rank of the ∈ M . At each point Jacobian matrix DX φ is equal to m at each point X X of a smooth manifold M , an (n − m) dimensional tangent space TX M is defined. 41

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42

3.1.2

Existence

Theorem 3.1 (Implicit function theorem). ∗ is non-singular, If the Jacobian matrix DX φ evaluated at a point X then there is a smooth locally defined function xn = ϕ (x1 , x2 , ..., xn−1 ), ϕ : Rn → Rm such that φ [x1 , x2 , ..., xn−1 , ϕ (x2 , x2 , ..., xn−1 )] = 0 for in some neighborhood of X ∗ ∈ Rn . So, there exists a set M = all X {X : φ(X) = 0} such that M is a compact manifold given by the graph of a t C ∞ function xn = ϕ (x1 , x2 , ..., xn−1 ) for [x1 , x2 , ..., xn−1 ] ∈ E ⊂ Rn . - Two-dimensional dynamical systems: φ (x, y) = 0 where φ : R2 → Rm = [x, y]t ∈ E ⊂ R2 . So, the implicit function for some m ≤ 2 and with X ∗ theorem implies that if the Jacobian matrix DX φ evaluated at a point X is non-singular, then there is a smooth locally defined function y = ϕ (x), in some neighborhood of ϕ : R2 → R1 such that φ [x, ϕ (y)] = 0 for all X ∗ 2 X ∈R - Three-dimensional dynamical systems: φ (x, y, z) = 0 where φ : R3 → = [x, y, z]t ∈ E ⊂ R3 . So, the implicit Rm for some m ≤ 3 and with X function theorem implies that if the Jacobian matrix DX φ evaluated at a ∗ is non-singular, then there is a smooth locally defined function point X in some z = ϕ (x, y), ϕ : R3 → R1 such that φ (x, y, ϕ (x, y)) = 0 for all X ∗ 3 neighborhood of X ∈ R

3.2

Invariant sets

3.2.1

Global invariance

Definition 3.1. A global invariant set S for a flow Φt : D → Rn associated with dynamical system (2.1) is defined as a subset S ⊂ Rn such that ∈S Φt (X)

∈S ∀ X

for all t ∈ R

(3.2)

Example 3.1. Global invariant sets Fixed points and periodic orbits are global invariant sets.

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43

Let’s consider a n-dimensional linear dynamical system (2.1): dX X) = JX = ( dt where J is a constant Jacobian matrix. A general solution to (2.1) may be obtained by linear superposition of n linearly independent solutions. If J has n linearly independent (generalized ) eigenvectors1 certain solutions lie in the linear subspaces spanned by the eigenvectors and defined as: − W S = span Y λi λ0 W C = span Y j

(3.3)

+ W U = span Y λk with i = 1, . . . , nS , j = 1, . . . , nC and k = 1, . . . , nU . W S , W C and W U represent respectively stable, center and unstable − , Y 0 and Y + are the nS , nC and globally invariant subspaces where Y λi λi λi nU (generalized) eigenvectors whose eigenvalues have negative, zero and ∗ , these global invariant positive real parts, respectively. For a fixed point X sets may also read: : Φt (X) ∗) = X →X ∗ , t → +∞ W S (X ∗) = X : Φt (X) →X ∗ , t → −∞ W U (X

1 For

generalized see Braun (1978) for example.

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44

3.2.2

DGeometry

Local invariance

Definition 3.2. A local invariant set S for a flow Φt : D → Rn associated with dynamical system (2.1) is defined as a subset S ⊂ Rn such that ∈S Φt (X)

∈S ∀ X

for

|t| < T

with

T >0

(3.4)

Let’s suppose that the n-dimensional nonlinear dynamical system (2.1) ∗ ) and ∗ . Then, intersections of W S (X has a hyperbolic fixed point X h h ∗ ) with a sufficiently small neighborhood of X ∗ contain smooth2 W U (X h h S ∗ U ∗ local invariant manifolds Wloc (X ) and Wloc (X ) defined as: S ∗ : Φt (X) →X ∗ , t → +∞ Wloc (Xh ) = X h U ∗ : Φt (X) →X h∗ , t → −∞ (Xh ) = X Wloc S U Invariant manifolds Wloc and Wloc are nonlinear analogues of the stable S and and unstable W S and W U eigenspaces. Next Th. 3.2 states that Wloc U S U ∗ Wloc are tangent to W and W at Xh .

Theorem 3.2. Stable Manifold Theorem for a Fixed Point ∗. Suppose that dynamical system (2.1) has a hyperbolic fixed point X h S ∗ U ∗ Then, there exist local stable and unstable manifolds Wloc (Xh ), Wloc (Xh ) of the same dimensions as those of the eigenspaces W S , W U of linearized ∗ . W S (X ∗ ) and W U (X ∗ ) are system (2.2), and tangent to W S , W U at X h loc h loc h X) is. as smooth as the function ( Proof.

Cf. Carr (1981, p. 15)

2 differentiable

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MF 10

45

Example 3.2. Stable and unstable manifolds Let’s consider the system (Guckenheimer and Holmes (1983, p. 14)): dX dt

dx

dt dy dt

=

f (x, y)

g (x, y)

=

x

−y + x2

The functional Jacobian matrix evaluated at the unique hyperbolic fixed ∗ (0, 0) reads: point X h ∗ = 1 0 J X with the eigenvalues λ1 = 1 and λ2 = −1. 0 −1 X=

∗

X h

The linearized system has two invariant subspaces: ! " − = (x, y) ∈ R 2 x = 0 W S = span Y λ2 ! " + = (x, y) ∈ R 2 y = 0 W U = span Y λ1 Although its nonlinear feature, this system may be integrated exactly (Cf. Ch. 6, Ex. 6.2):

y=

x2 C + 3 x

where C is a constant depending on initial conditions. According to HartU ∗ (Xh ) can be man-Grobman theorem and to the previous definitions Wloc U is represented by the graph y = h (x) with h (0) = h (0) = 0, since Wloc ∗ (0, 0). Thus, C = 0 and we obtain (Cf. 3.1) tangent to W U at X h ∗) = W U (X h

# x2 (x, y) ∈ R 2 y = 3

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46

Y

4

2

0

X

2

4

4

Fig. 3.1

2

0

2

4

Stable W S and unstable W U manifolds in green and blue respectively.

Let’s notice that in this example W U is the global and not local manifold. This will be stated in Ch. 6, Ex. 6.6. Remark. There have been many applications of the search for invariant manifolds in dynamical systems such as the Centre Manifold Theory or the Geometric Singular Perturbation Theory which endeavor to identify the central dynamical structures (invariant sets, invariant manifolds) in singularly perturbed systems (Cf. supra).

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DGeometry

Chapter 4

Local Bifurcations

“Mathematicians do not study objects, but relations between objects.” — H. Poincaré —

Bifurcation in a set of differential equations, i.e. in autonomous dynamical systems is concerned with changes in the qualitative behavior of the corresponding phase portrait as parameters vary and more precisely, when such a bifurcation parameter reaches a certain value, called critical value. Thus, bifurcation theory is of great importance in dynamical systems study because it indicates stability changes, structural changes in a system etc. So, plotting the solution of autonomous dynamical system according to the bifurcation parameter leads to the construction of a bifurcation diagram. Such diagram provides knowledge on the behavior of the solution: constant, periodic, nonperiodic or even chaotic as pointed out by Hoppensteadt and Hyman (1977). 4.1

Center Manifold Theorem

Let’s consider a system of differential equations defined in a compact E in = [x1 , x2 , ..., xn ]t ∈ ∈ Rp with X cluded in Rn depending on a parameter µ n E⊂R : dX X, µ) = ( dt

(4.1)

t X, µ) = f1 (X, µ), f2 (X, µ), ..., fn (X, µ) ∈ E ⊂ Rn defines in E where ( a velocity vector field whose components fi , supposed to be C ∞ continuous 47

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48

functions in E with values in R, are checking the assumptions of the CauchyLipshitz theorem. According to Guckenheimer and Holmes (1983, pp. 123-125) the center manifold theorem recalled below provides “a means for systematically reducing the dimension of the state spaces which need to be considered when analyzing bifurcations of a given type.” 4.1.1

Center manifold theorem for flows

∗ (i.e. Let suppose dynamical system (4.1) has a nonhyperbolic fixed point X there are eigenvalues with zero real part, say nC ) at the origin such that the spectrum of the functional jacobian matrix may divided in three parts:   < 0 if λ ∈ W S Re (λ) = 0 if λ ∈ W C  > 0 if λ ∈ W U S Then there exist C r stable and unstable invariant manifolds Wloc and S U r−1 tangent to W and W at the fixed point and a C nC -dimensional C S tangent to W C at the fixed point. The manifolds Wloc , center manifold Wloc U C Wloc and Wloc are all invariant with respect to the flow Φt (X) of dynamical system (4.1). The stable and unstable manifolds are unique but W C needs not to be. U Wloc

Proof.

Cf. Carr (1981, p. 3) ; Guckenheimer and Holmes (1983, p. 127).

MF 11

Example 4.1. Nonuniqueness of the center manifold (Kelley, 1967) Let’s consider the following dynamical system: V

2 x˙ f (x, y) x = = y˙ g (x, y) −y

Although its nonlinear feature this system may be easily integrated since it has an integrating factor (Cf. Ex. 6.1). Its solution curves may be represented (Cf. Fig. 4.1) by the graph of the functions: 1

y (x) = ce x

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49

For x < 0, there is an infinite number of these solution curves approaching the origin with dy/dx = 0. While for x 0, the only solution curve which approaches the origin is the x -axis1 . So, the center manifold is not unique. Y 4

2

0

X

2

4 4

Fig. 4.1

2

0

2

4

Part of the center manifold in green

4.1.2

Center manifold approximation

Let’s rewrite2 dynamical system (4.1) as:

x˙ = Ax + f (x, y ) y˙ = By + g (x, y)

(4.2)

where x ∈ Rn , y ∈ Rm and A is a n × n constant matrix such that all its eigenvalues have zero real parts while B is a m × m constant matrix such that all its eigenvalues have negative real parts. The functions f and g are C 2 with f(0, 0) = 0, DX f(0, 0) = 0, g(0, 0) = 0, DX g(0, 0) = 0 and x, y ) so that D represents the functional jacobian matrix. When the X( X functional Jacobian matrix is diagonalizable dynamical system (4.1) may be transformed into (4.2) while using the eigenbasis. 1 It

will be stated in Ex. 9.1 that the x-axis is invariant. sake of simplicity arrows could be forgotten.

2 For

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50

C Locally the center manifold Wloc may be represented in the vicinity of ∗ at the origin as: the fixed point X

C ∗ Wloc (X = 0) = (x, y ) ∈ Rn × Rm | y = h(x), |x| < δ, h(0) = 0, D xh(0) = 0 for δ small enough. Conditions h(0) = 0 and D xh(0) = 0 imply that C ∗ (X = 0) is tangent to W C at (x, y ) = (0, 0). Wloc Thus, by plugging y = h(x) into the first equation of (4.2) leads to: x˙ = Ax + f(x, h(x)),

x ∈ Rn ,

h : Rn → Rm

(4.3)

Then, since h(x) is tangent to y = 0 the following theorem enables to link the dynamics of system (4.2) to that of system (4.3). Theorem 4.1 (Henry, Carr). If the origin x = 0 of (4.3) is locally asymptotically stable (resp. unstable) the origin of (4.2) is also locally asymptotically stable (resp. unstable). Proof.

Cf. Henry (1981) ; Carr (1981, p. 14 and next).

Thus, for the computation of y = h(x) let’s replace in (4.2) y˙ by:

y˙ = D xh(x)x˙ ⇔ D xh(x) Ax + f(x, h(x)) = Bh(x) + g (x, h(x))

(4.4)

According to Guckenheimer and Holmes (1983, p. 131), “this (partial) differential equation for h (x) cannot be solved exactly. So, its solution can be approximated arbitrarily closely as a Taylor series at x = 0.” Example 4.2. Center manifold 2D Let’s consider the following dynamical system (Guckenheimer and Holmes, 1983, p. 133)

V

x˙ xy f (x, y) = = y˙ g (x, y) −y + αx2

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The functional Jacobian matrix computed at the fixed point, i.e. the origin provides two eigenvalues: 0 and −1. So , there exists a center manifold solution of Eq. (4.4) which may be approximated arbitrarily closely as a Taylor series at x = 0 by: y = h (x) = a20 x2 + a30 x3 + a40 x4 + a50 x5 + O x6 By substituting y into Eq. (4.4) which transcribes the invariance of the center manifold we have: y˙ = h (x) x˙

−h (x) + αx2 = h (x) (xh (x))

⇔

Solving this equation order by order Order Order Order Order

x2 : 3

x : 4

x : 5

x :

−a20 + α

=0

a30

=0

a40 +

2a220

8a20 a30 − a50

=0 =0

provides the unknown coefficients: a20 = α, a30 = a05 = 0, a40 = −2α2 Thus, the center manifold reads: y = h (x) = αx2 − 2α2 x4 + O x6 and the reduced system is x˙ = αx3 − 2α2 x5 + O x7 Example 4.3. Center manifold 3D Let’s consider the dynamical system (Dang-Vu, 2000, p. 40):       x˙ f (x, y, z) −y + xz  g (x, y, z)  =   y˙  =  V x2+ yz2 2 z˙ h (x, y, z) −z − x + y + z

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The center manifold z = h (x, y) solution to the partial equation (4.4) may be approximated arbitrarily closely as a Taylor series at (x, y) = (0, 0): z = h (x, y) = a20 x2 + a11 xy + a02 y 2 + a30 x3 + a21 x2 y + a12 xy 2 + a03 y 3 + · · · By substituting z into Eq. (4.4) which transcribes the invariance of the center manifold we have:

z˙ =

∂h ∂h x˙ + y˙ ∂x ∂y

Replacing z by h(x, y) into the previous equation leads to: ∂h ∂h (−y + xh(x, y)) + (x + yh(x, y)) = −h(x, y) − (x2 + y 2 ) + h2 (x, y) ∂x ∂y Solving this equation order by order Order

x2 :

a20 + a11 + 1

=0

Order

xy :

a20 − a11 − 2a02

=0

a11 − a02 − 1

=0

a30 + a21

=0

Order Order Order Order Order

2

y : 3

x : 2

x y: 2

xy : 3

y :

a21 − 3a30 + 2a12

=0

a12 − 2a21 + 3a03

=0

a03 − a12

=0

provides the unknown coefficients: a20 = a02 = −1, a11 = a30 = a03 = a21 = a12 = 0 Thus, the center manifold equation reads: z = h (x, y) = −x2 − y 2 + O x4 , y 4

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4.1.3

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53

Center manifold depending upon a parameter

Dynamical system (4.1) is indeed depending upon a vector of parameters µ. Thus, system (4.2) may be rewritten as:    x˙ = Ax + f (x, y, µ) y˙ = By + g (x, y , µ)   µ˙ = 0

(4.5)

with (x, y, µ) ∈ Rn × Rm × Rp . The (n + p)-dimensional center manifold reads: y = h (x, µ ) and satisfies the following (partial) differential equations system:

) Ax + f(x, h(x, µ )) = Bh(x, µ ) + g(x, h(x, µ )) D xh(x, µ

(4.6)

The center manifold will be very useful for detecting local bifurcations of codimension 1. Example 4.4. Center manifold depending upon a parameter. Let’s consider the dynamical system (Carr, 1981, p. 7):       x˙ f (x, y, µ) µx − x3 + xy  g (x, y, µ)  = −y + y 2 − x2  V  y˙  = 0 µ˙ h (x, y, µ) The functional Jacobian matrix computed at the fixed point, i.e. the origin provides two eigenvalues: µ and −1. So, when µ → 0 there exists a center manifold solution of Eq. (4.6) which may be approximated arbitrarily closely as a Taylor series at x = 0 by: y = h (x, µ) = a20 x2 + a11 µx + a02 µ2 + O x3 , µ3 By substituting y into Eq. (4.6) gives: y˙ =

∂h (x) x˙ ∂x

⇔

2 −h (x) + (h (x)) − x2 = h (x) µx − x3 + xh (x)

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Solving this equation order by order provides the unknown coefficients: a20 = −1, a11 = a02 = 0 Thus, the center manifold reads: y = h (x, µ) = −x2 + O x3 , µ3 and the reduced system is x˙ = µx − 2x3 + O x4 , µ4 This dynamical system exhibits a pitchfork bifurcation (Cf. supra). 4.2

Normal Form Theorem

According to Wiggins (1990, p. 193 and next): “for simplifying dynamical systems, two approaches come to mind: one, reduce the dimensionality of the system and two, eliminate the nonlinearity. Two rigorous mathematical techniques that allow substantial progress along both lines of approach are centre manifold theory and the method of normal forms.” These techniques are the most important, generally applicable methods available in the local theory of dynamical systems.” The aim of normal forms computations is to build a sequence of nonlinear transformation (variable changes) which successively removes the nonlinear terms starting from the quadratics ones in order to reduce the vector field to its linear part. Of course, needless to say that this is not always possible and one has to be satisfied with the “simplest” form called normal form which may not be unique. Let’s suppose that the center manifold approximation has been applied to dynamical system (4.1) which has been transformed into (4.5). While omitting explicit reference to parameters µ and to the center manifold y = h (x, µ ) such system may be written as: x˙ = F (x) = Ax + f (x)

(4.7)

t t x = [x1 , x2 , . . . , xn ] ∈ Rn , f (x) = [f1 (x) , f2 (x) , . . . , fn (x)] ∈ E ⊂ Rn and A = D x F (0). The Taylor series of f (x) at the origin leads to: k+1 x˙ = Ax + F (2) (x) + F (3) (x) + · · · + F (k) (x) + O(|x| )

(4.8)

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(k)

t (k) (k) F (k) (x) = F1 (x) , F2 (x) , . . . , Fn (x) are homogeneous polynomial of degree k in x. Let’s pose L = Ax, then L induces a map ad L in the linear space Hk defined by: ad L (Y ) = [Y, L] = (DL) Y − (DY ) L

(4.9)

where DL = A and [·, ·] denotes the Lie bracket operation (Abraham and Marsden, 1987; Choquet-Bruhat et al., 1977). The component form reads:

[Y, L]i =

n ∂Li j=1

∂xj

Yj −

∂Yi Lj ∂xj

(4.10)

Theorem 4.2 (Poincar´ e, 1879). By a sequence of analytic coordinate changes of the form: x = y + P (y ) ,

P (y) ∈ Hr ,

r = 2, 3, . . . , k

dynamical system (4.8) is transformed into the normal form: k+1 ) y˙ = Ay + g (2) (y ) + g (3) (y) + · · · + g (k) (y ) + O(|y |

with g (i) ∈ Gi , 2 ≤ i ≤ k and where Gk is the complement for ad L (Hk ) in Hk such that Hk = ad L (Hk ) ⊕ Gk Proof. Cf. Guckenheimer and Holmes (1983, p. 141) ; Wiggins (1990, p. 212) ; Dang-Vu (2000, p. 47). Assume that (4.8) has been transformed so that the terms of degree smaller than k − 1 lie in the complementary subspace Gi , 2 ≤ i ≤ k − 1. Then, let’s introduce a coordinate change of the form: x = y + P (y ) t where P (y) = [P1 (y ) , P2 (y ) , . . . , Pn (y )] are homogeneous polynomial of degree k − 1 with P (0) = 0 whose coefficients are to be determined. (I + D y P (y ))y˙ = A(y + P (y )) + g (2) (y ) + · · · k+1 ) · · · + g (k−1) (y ) + F (k) (y ) + O(|y|

(4.11)

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Multiplying Eq. (4.11) by (I + D y P (y ))−1 , i.e. by I − D y P (y) + h.o.t leads to: y˙ = Ay + g (2) (y ) + · · · + g (k−1) (y ) + F (k) (y ) k+1 +AP (y) − D y P (y) Ay + O(|y | )

(4.12)

Since L (y ) = Ay and A = D y L, Eq. (4.12) may be written: y˙ = Ay + g (2) (y) + · · · + g (k−1) (y) + F (k) (y ) k+1 + (D y L) P (y ) − (D y P (y))L (y ) + O(|y | )

(4.13)

According to (4.9), Eq. (4.13) reads: k+1 y˙ = Ay +g (2) (y )+· · ·+g (k−1) (y )+F (k) (y )+ad L(P (y))+O(|y | ) (4.14)

So the terms of degree smaller than k are unchanged by this transformation while the terms of degree k read: F (k) (y) + (D y L) P (y ) − (D y P (y))L = F (k) (y ) + ad L(P (y )) A suitable choice of P (y ) will make F (k) (y ) + ad L(P (y)) lie in Gk−1 , i.e. F (k) (y) can be eliminated if P (y) can be chosen such that: ad L(P (y )) = (D y L) P (y ) − (D y P (y))L = −F (k) (y )

(4.15)

This is known as the homological equation associated with the linear vector field Ay . Then, if the operator ad L(P (y )) is inversible a homogeneous polynomial P (y ) of degree k can be found (Fredholm alternative). Following Arrowsmith and Place (1990, p. 75 and next), the Lie bracket ad L(P (y)) : H k → H k is a map and its eigenvalues may be expressed in terms of those of A. If A has distinct eigenvalues λi , i = 1, . . . , n, its eigenvectors ei form a basis for Rn . Let x1 , . . . , xn be the coordinates of x relative

mn 1 m2 = xm (m1 , . . . , mn ). to this eigenbasis and let’s pose xm 1 x2 . . . xn with m k The matrix representing ad L(P (y )) in H is also diagonal and according to (4.9)

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ad L xm ei = (DL) xm ei − D xm ei L n mj m

ei − λi xj x ei = λi xm x j j=1

= λi − m. λ xm ei

(4.16)

ei is an eigenvector of ad L(P (y)) with eigenvalues where the monomial xm λ and λ (λ1 , . . . , λn ). Λm,i

= λi − m.

= λi − m. λ = 0 the compoThus in the non-resonant case, i.e. if Λm,i

nents of the homogeneous polynomial P (y) are given by: (k)

pm,i

=

Fm,i

m. λ − λi

(4.17)

k where pm,i are the coordinates of P (y) and Fm,i those of F (k) which can

be also expressed in the eigenbasis as following:

F (k) (x) = $

(k)

Fm,i xm ei

m,i

mj =k

According to Wiggins (1990, p. 193 and next), both techniques of centre manifold theory and the method of normal forms are the basis of development of bifurcation theory. Example 4.5. Normal form Consider the dynamical system (Arrowsmith and Place, 1990, p. 75): x˙ =

x˙1 x˙2

= F

f1 (x1 , x2 ) f2 (x1 , x2 )

3x1 − x22 = x2

The functional Jacobian matrix at the origin reads:

J(0) = A =

3 0

0 1

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So, the dynamical system may be written: x˙ = Ax + F (2) (x)

with

F (2) (x) =

2 −x2 0

A has two distinct real eigenvalues λ1 = 3 and λ2 = 1 for which the corresponding eigenvectors form a basis for R2 . 1 e1 = 0

and

0 e2 = 1

So, in this basis F (2) reads:

F (2) (x) = $

(k)

(2)

(2)

m1 m2 m1 m2

Fm,i xm ei = Fm,1 e1 + Fm,2 e2

x1 x2

x1 x2

m,i

mj =2

m1 m2 2 x1 x2 0 −x2 (2) (2) F (2) (x) = Fm,1 = + F 1 m2

m,2

0 x 0 xm 1 2 It follows that: 0 m = 2

and

(2) Fm,i

=

−1

if i = 1

0

if i = 2

So, according to Eq. (4.17) the components of the homogeneous polynomial P (y ) are given by: = 1 and pm,2 =0 pm,1

2 y2 Thus, P (y ) = and the coordinate changes x = y + P (y) reads: 0 2 y1 y2 x1 = + x2 y2 0 It may be checked that the dynamical system is transformed into the 3y 1 linear vector field y˙ = in which the quadratic terms have been y2 removed.

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Example 4.6. Normal form Let’s consider the dynamical system (Lynch, 2007, p. 135): x˙ =

x˙1 x˙2

= F

f1 (x1 , x2 ) f2 (x1 , x2 )

=

λ1 x1 + a20 x21 + a11 x1 x2 + a02 x22 λ2 x2 + b20 x21 + b11 x1 x2 + b02 x22

The functional Jacobian matrix at the origin reads: J(0) = A =

λ1 0

0 λ2

So, the dynamical system may be written: x˙ = Ax + F (2) (x)

with

F (2) (x) =

a20 x21 + a11 x1 x2 + a02 x22 b20 x21 + b11 x1 x2 + b02 x22

According to Eq. (4.17) components of the homogeneous polynomial P (y) =

f20 y12 + f11 y1 y2 + f02 y22 g20 y12 + g11 y1 y2 + g02 y22

are given by: a20 λ1 b20 = 2λ1 − λ2

a11 λ2 b11 = λ1

a02 2λ2 − λ1 b02 = λ2

f20 =

f11 =

f02 =

g20

g11

g02

(4.18)

So, if A has two distinct eigenvalues, i.e. if there is no resonance all of the quadratic terms can be eliminated from this dynamical system which can be transformed into a linear normal form. Then, normal form of example 4.5 may be directly found will using the previous formulae.

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4.3

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Local Bifurcations of Codimension 1

Let’s consider that the differential equations system (4.1) defined in a compact E included in R depends on a single parameter µ ∈ R: dX X, µ) = ( dt Let’s make the following assumptions: ∗ = X ∗ (µ) = X ∗ (0) = 0, • at µ = 0 the system (4.1) has a fixed point X • the functional Jacobian matrix associated with system (4.1) has a simple eigenvalue on the imaginary axis, i.e. λ1 = 0, ns eigenvalues with negative real part and nu eigenvalues with positive real part ; ns + nu + 1 = n. According to Center Manifold Theorem dynamical system (4.1) reads:    x˙ = A (µ) x + f (x, y , z, µ) y˙ = B (µ) y + g (x, y, z, µ)   z˙ = C (µ) z + h (x, y , z , µ)

(4.19)

where x ∈ Rn , y ∈ Rm , z ∈ Rp , f, g and h are nonlinear functions and A is a n × n constant matrix such that all its eigenvalues have zero real parts, B is a m × m constant matrix such that all its eigenvalues have negative real parts while C is a p × p constant matrix such that all its eigenvalues have positive real parts. In order to describe the dynamics of the system in the vicinity of (x, y , z , µ) = (0, 0, 0, 0), the following equation may be added to system (4.19): µ˙ = 0

(4.20)

Thus, for finding the center manifold associated with the generalized system (4.7)-(4.8) let’s pose:

y = hs (x, µ) z = hu (x, µ)

(4.21)

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where hs (x, µ) and hu (x, µ) are invariant manifolds satisfying:

y˙ = D xhs (x, µ) x˙ z˙ = D xhu (x, µ) x˙

(4.22)

Plugging hs (x, µ) and hu (x, µ) in Eq. (4.22) and while replacing x˙ , y˙ and z˙ leads to : 

  B (µ) hs + g(x, hs , hu , µ) = D xhs (x, µ) A (µ) x + f(x, hs , hu , µ)

  C (µ) hu + h(x, hs , hu , µ) = D xhu (x, µ) A (µ) x + f(x, hs , hu , µ) The solution of this system can be approximated arbitrarily closely as a Taylor series of hs (x, µ) and hu (x, µ) at (x, y , z , µ) = (0, 0, 0, 0). Then, system (4.7) reduces to

x˙ = A (µ) x + f(x, hs (x, µ) , hu (x, µ) , µ) = F (x, µ) µ˙ = 0

(4.23)

But since the functional Jacobian matrix associated with system (4.1) has a simple eigenvalue on the imaginary axis, i.e. with zero real part the following conditions may be added to system (4.23): F (0, 0) = D x F (0, 0) = 0

(4.24)

For a three-dimensional dynamical system for which (x, y , z , µ) = (x, y, z, µ) such conditions mean that the center manifold (a curve in this case) is tangent to the x-axis at the origin. Then, the Taylor series of F (x, µ) = F (x, µ) at (x, µ) = (0, 0) reads: 2 ∂F ∂F 1 ∂F ∂F +µ +µ + +h.o.t. F (x, µ) = F (0, 0)+ x x ∂x ∂µ (0,0) 2! ∂x ∂µ (0,0) Taking into account conditions (4.24) leads to: F (x, µ) = µ Fµ (x, µ)|(0,0) +

& 1 % 2 x Fxx + 2xµFxµ + µ2 Fµµ (0,0) + · · · 2! (4.25)

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Thus, the reduced system (4.23) may be written as:

x˙ = µ Fµ (x, µ)|(0,0) + µ˙ = 0

1 2!

%

x2 Fxx + 2xµFxµ + µ2 Fµµ

& (0,0)

+ ···

(4.26)

According to conditions upon the partial derivatives of F (x, µ) a saddlenode, a transcritical or a pitchfork bifurcation will occur. 4.3.1

Saddle-node bifurcation

Provided that the following conditions are checked: F (0, 0) = Fx (0, 0) = 0 ; Fµ (0, 0) = 0 ; Fxx (0, 0) = 0

(4.27)

a saddle-node or tangential bifurcation occurs. The fixed points of the reduced system (4.26) may be found while solving: F (x, µ) = 0.

x∗±

2 (x, µ) − (2µF (x, µ) + F µ2 Fxµ µ µµ (x, µ)) Fxx (x, µ) = Fxx (x, µ) (0,0) ( −2µFµ (x, µ) ∼± Fxx (x, µ) −µFµ (x, µ) ±

'

(0,0)

Thus, if Fµ (x, µ) Fxx (x, µ)|(0,0) < 0 there are two fixed points if µ > 0 and no fixed point if µ < 0. If Fµ (x, µ) Fxx (x, µ)|(0,0) > 0 there are two fixed points if µ < 0 and no fixed point if µ > 0. Their stability is given by the functional Jacobian matrix of the reduced system (4.26) which reads: J = Dx F (x, µ)|(0,0) = xFxx (x, µ) + µFxµ (x, µ)|(0,0) + h.o.t.

(4.28)

When µ → 0, the dominant term of the right hand side of Eq. (4.28) is xFxx (x, µ)|(0,0) . So, if Fxx (x, µ)|(0,0) < 0 fixed point x∗+ is stable and fixed point x∗− is unstable while if Fxx (x, µ)|(0,0) > 0 it is the reverse.

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Then, generically, as µ passes through µ = 0, the two fixed points collide, form a critical saddle-node equilibrium, and disappear. Example 4.7. Saddle-node bifurcation Let’s consider the dynamical system (Dang-Vu, 2000, p. 79):

x˙ = µ − x2 y˙ = −y

It may be easily checked that: F (0, 0) = Fx (0, 0) = 0

;

Fµ (0, 0) = 1 = 0 ;

Fxx (0, 0) = −2 = 0

So, this system exhibits a saddle-node bifurcation. 4.3.2

Transcritical bifurcation

Provided that the following conditions are checked: F (0, 0) = Fx (0, 0) = 0 ; Fµ (0, 0) = 0 ; Fxx (0, 0) = 0

(4.29)

a transcritical bifurcation occurs. Then, the reduced system (4.26) reads:

% & x˙ = 12! x2 Fxx (x, µ) + 2xµFxµ (x, µ) + µ2 Fµµ (x, µ) (0,0) + · · · µ˙ = 0

(4.30)

Fixed points of reduced system (4.30) are found by solving: F (x, µ) = 0. ) 2 (x, µ) − F −µFxµ (x, µ) ± µ Fxµ µµ (x, µ) Fxx (x, µ) ∗ x± = Fxx (x, µ)

(0,0)

2 Thus, if Fxµ (x, µ) − Fµµ (x, µ) Fxx (x, µ)(0,0) > 0 there are two fixed points x∗± .

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Their stability is given by the functional Jacobian matrix of the reduced system (4.30) which reads: J = Dx F (x, µ)|(0,0) = xFxx (x, µ) + µFxµ (x, µ)|(0,0) + · · ·

(4.31)

When µ → 0, the dominant term of the right hand side of Eq. (4.31) is still xFxx (x, µ)|(0,0) . So, if Fxx (x, µ)|(0,0) < 0 fixed point x∗+ is stable and fixed point x∗− is unstable while if. Then, generically, as µ passes through µ = 0, the real part of the equilibrium passes through zero. Both before and after the bifurcation, there is one unstable and one stable fixed point. However, their stability is exchanged when they collide. So the unstable fixed point becomes stable and vice versa. Example 4.8. Transcritical bifurcation Let’s consider the dynamical system (Dang-Vu, 2000, p. 80):

x˙ = µx − x2 y˙ = −y

It may be easily checked that: F (0, 0) = Fx (0, 0) = 0

;

Fµ (0, 0) = 0

;

Fxx (0, 0) = −2 = 0

So, this system exhibits a transcritical bifurcation. 4.3.3

Pitchfork bifurcation

Provided that the following conditions are checked: F (0, 0) = Fx (0, 0) = 0 ; Fµ (0, 0) = Fxx (0, 0) = 0 a pitchfork bifurcation occurs.

(4.32)

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Taking into account conditions (4.32) the Taylor series in the right hand side of Eq. (4.26) may be extended up to order 3. The first equation of the reduced system may be written: & 1 % 2 x Fxx + 2xµFxµ + µ2 Fµµ (0,0) 2! & 1 % 3 x Fxxx + 3x2 µFxxµ + 3xµ2 Fxµµ + µ3 Fµµµ (0,0) + · · · (4.33) + 3! √ While supposing that x ∼ O µ , the reduced system reads then:

x˙ = µ Fµ (x, µ)|(0,0) +

% x˙ = x µFxµ (x, µ) + µ˙ = 0

&

1 2 3! x Fxxx (x, µ) (0,0)

+ h.o.t.

(4.34)

Fixed points of reduced system (4.34) are found by solving: F (x, µ) = 0. ( x∗0

=0

;

x∗±

= ±

−6µFxµ (x, µ) Fxxx (x, µ)

(0,0)

If µFµx (x, µ) Fxxx (x, µ)|(0,0) < 0 there are three fixed points: x∗0 , x∗± . Their stability is given by the functional Jacobian matrix of the reduced system (4.34) which may be written:

J = Dx F (x, µ)|(0,0)

1 2 = µFxµ (x, µ) + x Fxxx (x, µ) + ··· 2 (0,0)

(4.35)

The functional Jacobian matrix evaluated at each fixed point reads: J (x∗0 ) = Dx F (x, µ)|(0,0) = µFxµ (x, µ)|(0,0) J x∗± = Dx F (x, µ)|(0,0) ∼ 2µFxµ (x, µ)|(0,0)

(4.36)

So, if Fµx (x, µ)|(0,0) > 0, Fxxx (x, µ)|(0,0) < 0 or Fµx (x, µ)|(0,0) < 0, Fxxx (x, µ)|(0,0) < 0 bifurcation is said subcritical and pitchfork is stable. If Fµx (x, µ)|(0,0) > 0, Fxxx (x, µ)|(0,0) > 0 or Fµx (x, µ)|(0,0) < 0, Fxxx (x, µ)|(0,0) > 0 bifurcation is said supercritical and pitchfork is unstable.

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Then there are intervals having a single stable fixed point and three fixed points (two of which are stable and one of which is unstable). Pitchfork bifurcations, like Hopf bifurcations have two types -supercritical or subcritical. In flows, that is, continuous dynamical systems described by ODEs, pitchfork bifurcations occur generically in systems with symmetry. Example 4.9. Pitchfork bifurcation Let’s consider the dynamical system (Dang-Vu, 2000, p. 83):

x˙ = µx − x3 y˙ = −y

It may be easily checked that: F (0, 0) = Fx (0, 0) = 0 ;

Fµ (0, 0) = 0 ;

Fxx (0, 0) = 0

So, this system exhibits a pitchfork bifurcation . 4.3.4

Hopf bifurcation

Let’s consider that the differential equations system (4.1) defined in a compact E included in R depends on a single parameter µ ∈ R: dX X, µ) = ( dt and let’s make the following assumptions: ∗ = X ∗ (µ) (i) one of the fixed points is depending on parameter µ, i.e. X (ii) the functional Jacobian matrix associated with this system has a pair of complex-conjugate eigenvalues λ1,2 = α (µ) ± iω (µ) such that: • for a critical value of the bifurcation parameter µ = µC : α (µC ) = 0 and

dα = 0 dµ µ=µC

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• the n − 2 others eigenvalues evaluated in µ have their real parts strictly negative. ∗ (µC ) , µC ) is a Hopf bifurIf these assumptions are checked, then (X cation point which leads to the appearance, from the equilibrium state ∗ (µC ), of a limit cycle. X For details see for example: Hopf (1942)3 ; Arnold (1974) ; Hassard and Wan (1978) ; Guckenheimer and Holmes (1983, pp. 151–152). Example 4.10. Hopf bifurcation Let’s consider the following dynamical system (Guckenheimer and Holmes, 1983, p. 146)

% 2 & 2 x˙ = −y + %x µ − 2x +2y& y˙ = x + y µ − x + y

which can be rewritten in polar coordinates as

r˙ = r µ − r2 θ˙ = 1

' where r = x2 + y 2 and θ = tan−1 (y/x). For µ < 0, the focus r = 0 is stable, but for µ > 0 the focus is unstable, and a stable circular limit cycle √ with r = µ and angular frequency ω = dθ/dt = 1 exists.

3 A translation of Hopf’s original paper may be found in Marsden and Cracken (1976, p. 163).

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Chapter 5

Slow-Fast Dynamical Systems

“Mathematics is the art of giving the same name to different things.” — H. Poincaré —

5.1

Introduction

The classical geometric theory developed originally by Andronov et al. (1937), Tikhonov (1948) and Levinson (1950) stated that singularly perturbed systems1 possess invariant manifolds on which trajectories evolve slowly and toward which nearby orbits contract exponentially in time (either forward and backward) in the normal directions. These manifolds have been called asymptotically stable (or unstable) slow manifolds. Then, Fenichel (1971, 1979) theory for the persistence of normally hyperbolic invariant manifolds enabled to establish the local invariance of slow manifolds that possess both expanding and contracting directions and which were labeled slow invariant manifolds. Various methods have been developed in order to determine the slow invariant manifold analytical equation associated with singularly perturbed systems. The seminal works of Wasow (1965), Cole (1968), O’Malley (1974, 1991) and Fenichel (1971, 1979) to name but a few, gave rise to the so-called Geometric Singular Perturbation Theory and the problem for finding the slow invariant manifold analytical equation turned into a regular perturbation problem.

1 For a Brief Historical Development of Singular Perturbations see O’Malley (1991, pp. 201-202).

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Dynamical system (2.1) comprising small multiplicative parameters in one or several components of its velocity vector field may be defined in a compact E included in R by:

x = f (x, z, ε) z = εg (x, z , ε)

(5.1)

where x ∈ Rm , z ∈ Rn , ε ∈ R+ and the prime denotes differentiation with respect to the independent variable t. The functions f and g are assumed to be C ∞ functions of x, z and ε in U × I, where U is an open subset of Rm × Rn and I is an open interval containing ε = 0. When 0 < ε 1, i.e. is a small positive number, variable x is called k fast variable, and z is called slow variable. Using Landau’s notation O ε represents a real polynomial in ε of k degree, with k ∈ Z. It is used to consider that generally x evolves at an O (1) rate; while z evolves at an O (ε) slow rate. Reformulating system (5.1) in terms of the rescaled variable τ = εt, we obtain the singularly perturbed systems:

εx˙ = f (x, z, ε) z˙ = g (x, z, ε)

(5.2)

Dots (·) represent the derivatives with respect to the new independent variable τ . The independent variables t and τ are referred to the fast and slow times, respectively, and (5.1) and (5.2) are called fast and slow system, respectively. These systems are equivalent whenever ε = 0, and they are labeled singular perturbation problems when ε 1, i.e. is a small positive parameter. The label singular stems in part from the discontinuous limiting behavior in system (5.1) as ε → 0+ . In such case, system (5.1) reduces to a m-dimensional system called reduced fast system, with the variable z as a constant parameter. System (5.2) leads to a differentialalgebraic system called reduced slow system which dimension decreases from m + n to n. By exploiting the decomposition into fast and slow reduced systems the geometric approach reduced the full singularly perturbed system to separate lower-dimensional regular perturbation problems in the fast and slow regimes, respectively. Geometric Singular Perturbation Theory is based on Fenichel’s assumptions (Fenichel, 1979) recalled below.

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Example 5.1. Paradigm of Relaxation Oscillator: Van der Pol model The oscillator of Balthazar Van der Pol (1926) is a second order system with nonlinear damping which may be written: x¨ + α(x2 − 1)x˙ + x = 0 The particular shape of the damping (which can can be modeled by an electronic circuit) makes decrease amplitude of the great oscillations and increase those of the small ones. According to D’Alembert (Cf. infra) the previous equation may be transformed into a system of two simultaneous equations of the first order.

x˙ = α(x + z − z˙ = − αx

x3 3 )

For large value of α, x becomes the “ fast ” variable and y the “ slow ” one. In order to analyze the limit α → ∞, a small parameter ε = 1/α2 as √ well as “ slow time ” t = t/α = t/ ε are introduced. Then, this singularly perturbed system reads:  3  εx˙ = f (x, z) = x + z − x 3  z˙ = g(x, z) = −x with ε a real positive parameter (usually ε = 0.05). Example 5.2. Paradigm of Complex Dynamics2 : Chua’s system The system of Leon Chua et al. (1986) is a relaxation oscillator with a cubic nonlinear characteristic elaborated from a circuit comprising a harmonic oscillator for which the operation is based on a field-effect transistor, coupled to a relaxation-oscillator composed of a tunnel diode. The modeling of the circuit uses a capacity which will prevent abrupt voltage drops 2 Cf.

Chua’s Circuit and the Qualitative Theory of Dynamical Systems, Mira, C. (1997); Development of the nonlinear dynamical systems theory from radio engineering to electronics, Letellier and Ginoux (2009)

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and makes it possible to describe the fast motion of this oscillator by the following equations which constitute a singularly perturbed system.  ˆ   εx˙ = f (x, y, z) = z − k(x) y˙ = g(x, y, z) = −z   z˙ = h(x, y, z) = −ax + y + bz The function kˆ (x) = c1 x3 +c2 x2 +µx describing the electrical response of the nonlinear resistor is an odd-symmetric function similar to the piecewise linear nonlinearity used in Chua’s piecewise linear model (Cf. supra) and for which the parameters are determined while using least-square method. Classical parameters set is: ε = 0.01, c1 = 44/3, c2 = 41/2, µ = 6.94, a = 0.7, b = 0.24. Remark. Both Van der Pol and Chua systems will be extensively studied in the second part of this book (Cf. supra).

5.2

Geometric Singular Perturbation Theory

Geometric Singular Perturbation Theory is based on the following assumptions and theorem stated by Nils Fenichel in the middle of the seventies3 . 5.2.1

Assumptions

(H1 ) Functions f and g are C ∞ functions in U × I, where U is an open subset of Rm × Rn and I is an open interval containing ε = 0. (H2 ) There exists a set M0 that is contained in {(x, z) : f (x, z, 0) = 0} such that M0 is a compact manifold with boundary and M0 is given 0 (z ) for z ∈ D, where D ⊆ Rn by the graph of a C 1 function x = X is a compact, simply connected domain and the boundary of D is an (n − 1) dimensional C ∞ submanifold. Finally, the set D is overflowing invariant with respect to (5.2) when ε = 0. 3 For

an introduction to Geometric Singular Perturbation Methods see Kaper (1999).

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(H3 ) M0 is normally hyperbolic relative to the reduced fast system and in particular it is required for all points p ∈ M0 , that there are k (resp. p, 0) with positive (resp. negative) real parts l) eigenvalues of D x f ( bounded away from zero, where k + l = m. Theorem 5.1. Fenichel’s persistence theorem Let system (5.1) satisfying the conditions (H1 ) − (H3 ). If ε > 0 is (z, ε) defined on D such sufficiently small, then there exists a function X that the manifold Mε = {(x, z ) : x = X (z, ε)} is locally invariant under (z, ε) is C r for any r < +∞, and Mε is C r O (ε) close (5.1). Moreover, X to M0 . In addition, there exist perturbed local stable and unstable manifolds of Mε . They are unions of invariant families of stable and unstable fibers of dimensions l and k, respectively, and they are C r O (ε) close for all r < +∞, to their counterparts. Proof.

5.2.2

Cf. Fenichel (1971) ; Jones (1994) ; Kaper (1999).

Invariance

Generally, Fenichel theory enables to turn the problem for explicitly finding (z, ε) whose graphs are locally slow invariant manifolds functions x = X Mε of system (5.1) into regular perturbation problem. Invariance of the (z , ε) satisfies: manifold Mε implies that X (z, ε) g X (z , ε) , z , ε = f X (z, ε) , z, ε εD z X

(5.3)

Then, plugging the perturbation expansion: (z, ε) = X 0 (z) + εX 1 (z ) + O ε2 X (z , ε). into (5.3) enables to solve order by order for X (z , ε) , z , ε up to terms of order The Taylor series expansion for f X two in ε leads at order ε0 to 0 (z ) , z, 0 = 0 f X

(5.4)

0 (z) due to the invertibility of D x f and the implicit function which defines X theorem.

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At order ε1 we have: 0g X 0 , z, 0 0 , z, 0 = D x f X 1 + ∂f X 0 , z, 0 X D z X ∂ε

(5.5)

1 (z) and so forth. which yields X 0g X 0 , z, 0 0 , z, 0 X 1 = D z X 0 , z , 0 − ∂ f X D x f X ∂ε

(5.6)

So, regular perturbation theory enables to build locally slow invariant manifolds Mε . But for high-dimensional singularly perturbed systems slow invariant manifold analytical equation determination leads to tedious calculations. In the second part of this work an alternative approach to the Geometric Singular Perturbation Theory will be introduced. Proof. For application of this technique cf. Fenichel (1979). Also, this procedure is similar to those used by Carr (1981) to compute the center manifold approximation presented in the previous Ch. 4. 5.2.3

Slow invariant manifold

(z, ε) satisAs previously stated invariance of manifold Mε implies that: X fies Eq. (5.3). The aim of this subsection is to provide an explicit formulae i (z) (with i = 0, 1, 2) of the perturbation expansion asfor the first orders X sociated with low-dimensional two and three singularly perturbed dynamical systems. MF 12

Two-dimensional dynamical systems First, let’s recall the Taylor series expansion for a function of three variables f (x, y, ε) in the neighborhood of the point x = x0 + n, y = y0 + p, ε = ε0 + q up to second-order terms in x, y and ε. 2 ∂f ∂f 1 ∂f ∂f ∂f ∂f +p +q f (x, y, ε) = f (x0 , y0 , ε0 )+n +p +q + +· · · n ∂x ∂y ∂ε 2! ∂x ∂y ∂ε 2 ∂f ∂f where n ∂f represents a “symbolic power” expanded for∂x + p ∂y + q ∂ε mally by the binomial formulae by taking into account that power two of the derivative corresponds to the second derivative and not to the square first derivative.

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According to the nature of the problem perturbation expansion reads: y = Y (x, ε) = Y0 (x) + εY1 (x) + ε2 Y2 (x) + O ε3 Thus, by posing:    x = x0 y = Y0 + p = Y0 + εY1 + ε2 Y2 + · · ·   ε0 = 0

⇔

  n = 0 p = Y0 + εY1 + ε2 Y2 + · · ·   q=ε

So, the Taylor series expansion may be written: ∂f ∂f +ε ∂y ∂ε 2 ∂f 1 ∂f +ε + + ··· (εY1 + ε2 Y2 ) 2! ∂y ∂ε

f (x, y, ε) = f (x, Y0 , 0) + (εY1 + ε2 Y2 )

If the function f is independent of ε it reduces to: f (x, y) = f (x, Y0 ) + (εY1 + ε2 Y2 )

2 ∂ 2 f 1 ∂f + εY1 + ε2 Y2 + · · · (5.7) ∂y 2! ∂y 2

Let’s consider the following singularly perturbed dynamical system:

εx˙ = f (x, y) y˙ = g(x, y)

According to Eq. (5.3) invariance of the manifold Mε reads:

∂Y ∂x

f (x, Y ) = εg (x, Y )

Plugging in this equation the perturbation expansion leads to:

∂Y0 ∂f ∂Y1 2 ∂Y2 +ε +ε f (x, Y0 ) + (εY1 + ε2 Y2 ) ∂x ∂x ∂x ∂y 2 2 ∂ f 1 + εY1 + ε2 Y2 2! ∂y 2 2 2 ε ∂ g ∂g 2 2 + = εg(x, Y0 ) + ε(εY1 + ε Y2 ) εY1 + ε Y2 ∂y 2! ∂y 2

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Then, solving order by order provides at: order ε0 : ∂Y0 f (x, Y0 (x)) = 0 ∂x

⇔

y = Y0 (x)

(5.8)

order ε1 : Y1 (x)

∂Y1 ∂Y0 ∂f + f (x, Y0 (x)) = g(x, Y0 (x)) ∂x ∂y ∂x

Since according to implicit function theorem f (x, Y0 (x)) = 0 we have: Y1 (x) =

g(x, Y0 (x)) ∂Y0 ∂f (x, Y0 (x)) ∂x ∂y

(5.9)

order ε2 :

Y2 (x)

∂Y0 ∂f 1 2 ∂Y0 ∂ 2 f ∂Y1 ∂f ∂Y2 ∂g + Y1 (x) + f (x, Y0 (x)) = Y1 (x) +Y1 (x) ∂x ∂y 2 ∂x ∂y 2 ∂x ∂y ∂x ∂y

Since according to implicit function theorem f (x, Y0 (x)) = 0 we have: Y1 (x) Y2 (x) =

∂g ∂Y0 ∂ 2 f ∂Y1 ∂f (x, Y0 (x)) − 12 Y12 (x) (x, Y0 (x)) (x, Y0 (x)) − Y1 (x) ∂y ∂x ∂y 2 ∂x ∂y ∂Y0 ∂f (x, Y0 (x)) ∂x ∂y

(5.10)

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77

Example 5.3. Van der Pol system  3  εx˙ = f (x, y) = x + y − x 3  y˙ = g(x, y) = −x Van der Pol system is checking Fenichel’s assumptions (H1 ) − (H3 ). So, its singular approximation M0 is contained in {(x, y) : f (x, y) = 0} such that M0 is a compact manifold with boundary given by the graph of the C 1 function: y = Y0 (x) = x3 /3 − x. Thus, the problem is to find a function y = Y (x, ε) whose graph is locally slow invariant manifold Mε of the Van der Pol system. Let’s pose: Y (x, ε) = Y0 (x) + εY1 (x) + ε2 Y2 (x) + O ε3 As previously stated order ε0 leads to ∂Y0 f (x, Y0 (x)) = 0 ∂x

⇔

Y0 (x) =

x3 −x 3

which defines the singular approximation Y0 (x) = x3 /3 − x due to the invertibility of ∂f /∂y and the implicit function theorem. ∂f ∂Y0 (x) (x, Y0 (x)) = 1 and = x2 − 1 we have ∂y ∂x according to Eq. (5.9) at order ε1 : Since g(x, Y0 (x)) = −x ,

Y1 (x) = At order ε2 since

x 1 − x2

∂2f 1 + x2 ∂g ∂Y1 = = = 0 and Eq. (5.10) gives: 2 ∂y ∂y ∂x (1 − x2 )2 Y2 (x) =

x(1 + x2 ) (1 − x2 )4

Slow manifold equation associated with the Van der Pol system reads: 2 x3 x 2x 1 + x y= −x+ε +ε + O ε3 4 2 3 1−x (1 − x2 )

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MF 13

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Three-dimensional dynamical systems First, let’s recall the Taylor series expansion for a function of four variables f (x, y, z, ε) in the neighborhood of the point x = x0 + n, y = y0 + p, z = z0 + q, ε = ε0 + r up to second-order terms in x, y, z and ε. ∂f ∂f ∂f ∂f +p +q +r ∂x ∂y ∂z ∂ε 2 ∂f 1 ∂f + ···+ r + n 2! ∂x ∂ε

f (x, y, z, ε) = f (x0 , y0 , z0 , ε0 ) + n

2 ∂f where n ∂f still represents a “symbolic power” expanded ∂x + · · · + r ∂ε formally by the binomial formulae. According to the nature of the problem perturbation expansion reads: z = Z(x, y, ε) = Z0 (x, y) + εZ1 (x, y) + ε2 Z2 (x, y) + O ε3 Thus, by posing:  x = x0     y = y 0 2  z = Z  0 + q = Z0 + εZ1 + ε Z2 + · · ·    ε=0

⇔

 n=0     p=0  q = Z0 + εZ1 + ε2 Z2 + · · ·     r=ε

So, the Taylor series expansion may be written: ∂f ∂f +ε f (x, y, z, ε) = f (x, y, Z0 , 0) + (εZ1 + ε2 Z2 ) ∂z ∂ε 2 1 ∂f ∂f + +ε (εZ1 + ε2 Z2 ) 2! ∂z ∂ε If the function f is independent of ε it reduces to: f (x, y, z) = f (x, y, Z0 ) + (εZ1 + ε2 Z2 )

2 ∂ 2 f 1 ∂f + εZ1 + ε2 Z2 (5.11) ∂z 2! ∂z 2

Let’s consider the following singularly perturbed dynamical system:    εx˙ = f (x, y, z) y˙ = g(x, y, z)   z˙ = h(x, y, z)

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According to Eq. (5.3) invariance of the manifold Mε reads: ∂Z ∂Z f (x, y, Z) + ε = εh (x, y, Z) ∂x ∂y Plugging in this equation the perturbation expansion leads to: ∂Z1 ∂Z2 ∂f ∂Z0 +ε + ε2 f (x, y, Z0 ) + (εZ1 + ε2 Z2 ) ∂x ∂x ∂x ∂z 2 2 ∂Z0 1 ∂Z1 ∂ f 2 2 ∂Z2 + +ε +ε + εZ1 + ε Z2 g(x, y, Z0 ) 2! ∂z 2 ∂y ∂y ∂y 2 2 1 ∂ g ∂g + (εZ1 + ε2 Z2 ) + εZ1 + ε2 Z2 ∂z 2! ∂z 2 2 2 ε ∂ h ∂h 2 2 + = εh(x, y, Z0 ) + ε(εZ1 + ε Z2 ) εZ1 + ε Z2 ∂z 2! ∂z 2 Then, solving order by order provides at: order ε0 : ∂Z0 f (x, y, Z0 (x)) = 0 ∂x

⇔

z = Z0 (x, y)

(5.12)

order ε1 : ∂Z1 ∂Z0 ∂f ∂Z0 + f (x, y, Z0 (x, y)) + g(x, y, Z0 (x, y)) ∂x ∂z ∂x ∂y = h(x, y, Z0 (x, y))

Z1 (x, y)

Since according to implicit function theorem f (x, y, Z0 (x, y)) = 0 we have: h(x, y, Z0 (x, y)) − Z1 (x, y) =

∂Z0 g(x, y, Z0 (x, y)) ∂y

∂Z0 ∂f (x, Z0 (x, y)) ∂x ∂z

order ε2 : 1 ∂Z0 ∂f ∂Z0 ∂ 2 f ∂Z1 ∂f + Z12 (x, y) + Z1 (x, y) ∂x ∂z 2 ∂x ∂z 2 ∂x ∂z ∂Z2 ∂Z0 ∂g f (x, y, Z0 (x, y)) + Z1 (x, y) + ∂x ∂y ∂z ∂Z1 ∂h + g(x, y, Z0 (x, y)) = Z1 (x, y) ∂y ∂z

Z2 (x, y)

(5.13)

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Still using implicit function theorem we have:

Z1 Z2 (x, y) =

∂h ∂Z0 ∂ 2 f ∂Z1 ∂f ∂Z0 ∂g ∂Z1 − Z1 − 12 Z12 − Z1 − g(x, y, Z0 ) ∂z ∂x ∂z 2 ∂x ∂z ∂y ∂z ∂y ∂Z0 ∂f (x, y, Z0 ) ∂x ∂y

(5.14) MF 13

Example 5.4. Chua’s system 3  2   εx˙ = f (x, y, z) = z − c1 x + c2 x + µx y˙ = g(x, y, z) = −z   z˙ = h(x, y, z) = −ax + y + bz Chua system which is checking Fenichel’s assumptions (H1 ) − (H3 ) the singular approximation M0 is contained in {(x, y, z) : f (x, y, z) = 0} such that M0 is a compact manifold with boundary given by the graph of the C 1 function: z = Z0 (x) = c1 x3 + c2 x2 + µx. So, the problem is to find a function z = Z(x, ε) whose graph is locally slow invariant manifold Mε of the Chua system. Let’s pose: Z(x, ε) = Z0 (x) + εZ1 (x) + ε2 Z2 (x) + O ε3 As previously stated order ε0 leads to ∂Z0 f (x, y, Z0 (x)) = 0 ∂x

⇔

z = Z0 (x, y) = c1 x3 + c2 x2 + µx

which defines the singular approximation Z0 (x) = c1 x3 + c2 x2 + µx due to the invertibility of ∂f /∂z and the implicit function theorem. ∂f ∂Z0 ∂Z0 = 0, = 1 and = 3c1 x2 + 2c2 x + µ we have according ∂y ∂z ∂x to Eq. (5.13) at order ε1 : Since

Z1 (x, y) =

−ax + y + bc1 x3 + bc2 x2 + bµx 3c1 x2 + 2c2 x + µ

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∂2f ∂g ∂f ∂Z1 ∂h ∂Z0 = b, = 0, = −1, = 1, = = 2 ∂z ∂z ∂y ∂z ∂z ∂y and g(x, y, Z0 ) = −Z0 Eq. (5.14) leads4 at order ε2 :

81

Since

∂Z0 ∂x

−1

−1 ∂Z0 ∂Z1 + Z0 ∂x ∂x 3c1 x2 + 2c2 x + µ

bZ1 − Z1 Z2 (x, y) =

The slow manifold equation associated with the Chua’s system reads: z = Z(x, y, ε) = c1 x3 +c2 x2 +µx+ε

−ax + y + bc1 x3 + bc2 x2 + bµx +O ε2 2 3c1 x + 2c2 x + µ

5.3

5.3.1

Slow-fast dynamical systems – Singularly perturbed systems Singularly perturbed systems

Definition 5.1. Dynamical system (5.2) may be written: dX X, ε) = ( dt

(5.15)

t X, = [εx, z, ε]t ∈ E ⊂ Rn and ( ε) = f(x, z, ε), g(x, z, ε) which where X defines in E a velocity vector field whose components fi , supposed to be C ∞ continuous functions in E with values in R, are checking the assumptions of the Cauchy-Lipshitz theorem and where 0 < ε 1. Both dynamical systems (5.2) and (5.15) are called singularly perturbed systems or slow-fast dynamical systems. 5.3.2

Slow-fast autonomous dynamical systems

It has been shown (Rossetto et al., 1998) that some non-singularly perturbed systems, i.e. dynamical systems without any small multiplicative parameter in one or several components of their velocity vector field and, consequently, 4 The

complete expression of Z2 (x, y) may be found in Mathematica file MF 13.

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without any singular approximation, can be considered as slow-fast and may possess a slow manifold whose equation will be provided in the second part of this work according to the Flow Curvature Method. Definition 5.2. A non-singularly perturbed dynamical system (2.1) defined in a compact E included in R may be considered as slow-fast if its functional Jacobian matrix has at least one “fast” eigenvalue, i.e. with the largest absolute value of the real part over a huge domain of the phase space. Example 5.5. Lorenz system Although the Lorenz’s system has no singular approximation, it has been numerically checked (Rossetto et al., 1998) that its functional jacobian matrix possesses at least a large and negative real eigenvalue in a large domain of the phase space. So, it can be considered as a slow fast dynamical system but not as a singularly perturbed system. Thus, Geometric Singular Perturbation Theory can not provide the slow invariant manifold associated with Lorenz system. In the second part of this book the slow manifold analytical equation of Lorenz system will be given by Flow Curvature Method. Theorem 5.2 (Rossetto et al. (1998)). Any singularly perturbed system is a slow-fast dynamical system. Proof. A necessary and sufficient condition for a rank n functional Jacobian matrix Mε to have one – and only one – “fast” eigenvalue is for one – and only one – row, or column of Mε to have ε−1 in factor (excepted in nongeneric cases). Let’s consider that the rank n functional Jacobian matrix Mε may be written as: 

ε−1 a11 ε−1 a12  a21 a22  Mε =  .. ..  . . an1

· · · ε−1 a1n ··· a2n .. .. . .

··· ···

    

(5.16)

ann

Let’s call λi with i = 1, 2, . . . , n the eigenvalues of the matrix Mε resulting from the resolution of the Cayley-Hamilton polynomial. The sum and product of these eigenvalues read:

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83

λi = λ1 + λ2 + . . . + λn = p

i=1

= T r [Mε ] =

n

aii = ε−1 a11 + a22 + . . . + ann

(5.17)

i=1 n *

λi = λ1 λ2 · · · λn = q = Det [Mε ] = ε−1 Det [Mε=1 ]

(5.18)

i=1

While reasoning by abduction let’s suppose that all eigenvalues are “fast”, i.e. may be expressed as a polynomial of valuation −1 in ε: λi = O ε−1 . As a consequence, the product of all eigenvalues would be a polynomial of valuation −n in ε what would be in contradiction with Eq. (5.18). Thus, if one eigenvalue is “fast” all the others are necessary “slow”, i.e. may be expressed as polynomial of valuation 0 in ε. So, since the functional Jacobian matrix associated with any singularly perturbed system (5.15) has its first row multiplied by ε−1 it has one “fast” eigenvalue. Then, the corresponding eigenmode associated with this “fast” eigenvalue is said: • “evanescent” if it is negative, • “dominant” if it is positive. All others eigenvalues are called “slow”.

Remark. Geometric Singular Perturbation Theory is entirely devoted to singularly perturbed system and provides their slow invariant manifold according to Fenichel’s theorem. But for high-dimensional singularly perturbed systems computations leading to the determination of their slow invariant manifold analytical equation become tedious. Moreover, in the case of non-singularly perturbed systems, like that of Lorenz for example, such theory can not provide its slow manifold. So, in the second part of this book an alternative method for determining the slow manifold analytical equation of singularly perturbed systems or slow-fast dynamical systems, i.e. non-singularly perturbed systems called Flow Curvature Method will be presented.

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Chapter 6

Integrability

“A system of differential equations is only more or less integrable.” — H. Poincaré —

Modeling natural phenomena as accurately as possible first involves describing elementary processes the evolution of which may be represented by nonlinear differential equations or dynamical systems and, secondly, implies providing information about the future of the observed phenomenon which leads to the particular problem of integrating these equations. In this aim, many methods have been developed the two main of which consist in solving differential equations either by series expansions (Newton, Leibniz, Euler, . . . ) or explicitly by “quadratures” (closed-form), i.e. while using a combination of elementary functions divided into algebraic and transcendental categories. The former, also known as perturbation expansions, fails in the vicinity of singular points as pointed out by Henri Poincaré in his memoirs (Poincaré, 1886) while for the later Joseph Liouville has stated in a note (Liouville, 1839) that the solution of any differential equation can not be necessary expressed in terms of elementary functions. Then, the notion of integrability was introduced to distinguish integrable systems from non-integrable.

6.1

Integrability conditions, integrating factor, multiplier

In the first part of this book, Sec. 1.2, it has been recalled that, according to D’Alembert transformation, a nth order differential equation can be transformed into a system of n simultaneous first-order equations, i.e. into a dynamical system. It will be established in this section that a total differ85

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ential equation, i.e. a differential 1-form provides another way of obtaining a dynamical system. Moreover, integrability conditions of such equation will be used to state about the integrability of the corresponding dynamical system. 6.1.1

Two-dimensional dynamical systems

Let’s consider a first order and first degree total differential equation involving two variables, x and y which represents a differential 1-form: dφ = P (x, y) dx + Q (x, y) dy = 0

(6.1)

where P and Q are supposed to be C ∞ continuous functions in E with values in R, checking the assumptions of the Cauchy-Lipschitz theorem. = φ (x, y) = C General integral of this differential equation reads: φ(X) and defines a family of plane curves depending on an arbitrary constant C. Necessary and sufficient condition of integrability of a differential 1-form: The differential 1-form dφ = P (x, y) dx + Q (x, y) dy = 0 is exact if: ∂P ∂Q = ∂y ∂x

(6.2)

= [Q (x, y) , −P (x, y)]t this equation reads Denoting the vector field: = 0 where ∇ = Div( ) represents the divergence of . · · ∇ Proof. The differential 1-form dφ = P (x, y) dx + Q (x, y) dy = 0 is exact if path-independent:

dφ = P (x, y) dx + Q (x, y) dy =

∂φ ∂φ dx + dy ∂x ∂y

∂φ ∂φ and Q (x, y) = , ∂x ∂y respectively and partial derivatives of P and Q read: It implies that P and Q are of the form P (x, y) =

∂P ∂2φ = ∂y ∂y∂x

and

∂Q ∂2φ = ∂x ∂x∂y

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87

Then, it follows from Clairaut’s theorem (Schwarz’s theorem) that second partial derivatives of a function commute: ∂Q ∂P = ∂y ∂x

Integrating factor Let’s consider the differential 1-form P (x, y) dx + Q (x, y) dy = 0 for which the integrability condition (6.2) is not satisfied. Thus, if µ (x, y) is a function such that µ (P dx + Qdy) is exact (total) differential, i.e. µ (P dx + Qdy) = dφ (x, y), then µ (x, y) is said to be an integrating factor of this differential 1-form (6.1) satisfying the relation: ∂ (µQ) ∂ (µP ) = ∂y ∂x

⇔

Q

∂µ ∂µ −P +µ ∂x ∂y

∂Q ∂P − ∂x ∂y

=0

= [Q (x, y) , −P (x, y)]t By using the Lie derivative1 of the vector field it reads: + T r (J) µ(X) =0 LV µ(X)

(6.3)

Remark. Symbol µ is used in reference to the works of Gaston Darboux which introduced this notation in his memoir Darboux (1878, p. 68). Example 6.1. Integrating factor Let’s consider the dynamical system used in Ex. 4.1 (Kelley, 1967) represented by the following differential 1-form: ydx + x2 dy = 0 for which integrability condition (6.2) is not satisfied. So this differential 1-form is = 1 which not exact but integrable. Thus, it may be checked that µ(X) x2 y 1 verify Eq. (6.3) is an integrating factor. Multiplied by 2 the differential x y 1-fom reads: dx dy =0 + x2 y 1 See

⇔

Appendix for Lie derivative definition.

d(−

1 + ln y) = 0 x

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So, the general integral of this system may be written: = φ (x, y) = − φ(X)

1 + ln y = C x

⇔

1

y = ce x

From differential equation to dynamical system t

t

= [Q (x, y) , −P (x, y)] = [f (x, y) , g (x, y)] . Let’s pose: The differential 1-form (6.1) may be transformed into a symmetrical form 2 leading to a two-dimensional dynamical system and conversely. dx dy = = dt f (x, y) g (x, y)

dX dt

⇔

dx

dt dy dt

=

f (x, y) g (x, y)

(6.4)

= φ (x, y) = C, is the General integral of this dynamical system, φ(X) general integral of the differential equation (6.1) and conversely.

Necessary and sufficient condition of integrability of a 2D-dynamical system: According to the definition of functional jacobian matrix (Sec. 2.6.1) the integrability condition (6.2) is equivalent to: T r (J) =

∂g ∂f + =0 ∂x ∂y

So, dynamical system (6.2) is integrable if: T r (J) = 0. Example 6.2. Integrable system Let’s consider the system of Ex. 3.2 (Guckenheimer and Holmes, 1983): dX dt

dx

dt dy dt

=

f (x, y) g (x, y)

=

x 2

−y + x

⇔

dy dx = = dt x −y + x2

The corresponding differential 1-form is: dφ = −xy + x2 dx−xdy = 0 Since integrability condition (6.2) is satisfied this total differential equation is exact. 2 See Nemytskii and Stepanov (1960, p. 35) ; Davis (1962, p. 17) ; Petrovski (1966, p. 91)

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89

−y + x2 dx − xdy = 0 ⇔ x2 dx − (ydx + xdy) = 0 ⇔ d

x3 − xy 3

=0

The general integral of this system which corresponds to the previous result obtained in Ex. 3.2 reads: x3 − xy = C 3

= φ (x, y) = φ(X)

⇔

y=

x2 C − 3 x

Three-dimensional dynamical systems

6.1.2

Let’s consider a first order and first degree total differential equation involving three variables, x, y and z which represents a differential 1-form: P (x, y, z) dx + Q (x, y, z) dy + R (x, y, z) dz = 0

(6.5)

where P , Q and R are supposed to be C ∞ continuous functions in E with values in R, checking the assumptions of the Cauchy-Lipschitz theorem. = φ (x, y, z) = C General integral of this differential equation is: φ(X) which defines a family of surfaces depending on an arbitrary constant C. Necessary and sufficient condition of integrability of a differential 1-form: The differential 1-form (6.5) is exact if: ∂Q ∂P = , ∂y ∂x

∂Q ∂R = , ∂z ∂y

∂R ∂P = ∂x ∂z

(6.6)

Then, the differential 1-form (6.5) is integrable iff: P

∂R ∂Q − ∂y ∂z

+Q

∂R ∂P − ∂z ∂x

+R

∂Q ∂P − ∂x ∂y

=0

(6.7)

= [P (x, y, z) , Q (x, y, z) , R (x, y, z)]t By denoting the vector field: · (∇ = 0 where ∇ = Rot( ) represents the × ) × Eq. (6.7) reads: 3 rotational of vector . 3 This formulation has been introduced by Joseph Bertrand in Trait´ e de calcul differentiel et de calcul integral, Gauthier-Villars, Paris (1864-1870), tome I, p. 220 and next.

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Proof.

Let’s suppose4 that differential 1-form (6.5) reads: dφ = P (x, y, z) dx + Q (x, y, z) dy + R (x, y, z) dz = 0

Then it follows that:

dφ = P (x, y, z) dx + Q (x, y, z) dy + R (x, y, z) dz ∂φ ∂φ ∂φ dx + dy + dz = ∂x ∂y ∂z This implies that P , Q and R are of the form:

∂φ ∂φ = µP , = µQ and ∂x ∂y

∂φ = µR with µ a multiplier 5 . So, partial derivatives of P , Q and R read: ∂z ∂P ∂µ ∂Q ∂µ ∂2φ =µ +P =µ +Q ∂y∂x ∂y ∂y ∂x ∂x ∂Q ∂µ ∂R ∂µ ∂ 2φ =µ +Q =µ +R ∂z∂y ∂z ∂z ∂y ∂y

(6.8)

∂2φ ∂R ∂µ ∂P ∂µ =µ +R =µ +P ∂x∂z ∂x ∂x ∂z ∂z Then it follows from Clairaut’s theorem (Schwarz’s theorem) that second partial derivatives of a function commute. So, multiplying Eqs. (6.8) respectively by R, P , Q and adding each one to each other we have: ∂Q ∂R ∂R ∂P ∂P ∂Q +P +Q +P +Q µ R =µ R ∂y ∂z ∂x ∂x ∂y ∂z and integrability condition is thus deduced from Eq. (6.9).

(6.9)

Remark. Let’s note that the condition for a total differential equation to be exact corresponds to a vector field of divergence free in dimension two and of rotational free in dimension three. 4 This proof is due to Leonard Euler, Inst. Calc. Int. 3 (1770), p. 1. See Ince (loc. cit.) p. 53 for details. 5 For dynamical systems of dimension higher than two, the terminology integrating factor is replaced by multiplier.

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Multiplier Let’s consider the differential 1-form (6.5) for which the conditions (6.6) for such differential 1-form to be exact are not valid while integrability condition (6.7) is satisfied. Thus, if µ (x, y, z) is a function such that µ (P dx + Qdy + Rdz) = dφ is exact or total differential, then µ (x, y, z) is said to be a multiplier for this differential 1-form (6.5) satisfying: ∂ ∂ ∂ (µP ) + (µQ) + (µR) = 0 ∂x ∂y ∂z ⇔ ∂µ ∂µ ∂µ P+ Q+ R+µ ∂x ∂y ∂z

∂P ∂Q ∂R + + ∂x ∂y ∂z

=0

+ T r (J) µ(X) =0 which may also be written as previously: LV µ(X) Remark. Provided that the differential 1-form (6.2) (resp. 6.5) has one and only one solution it may be proved, e.g. Davis (1962, p. 35), that there exists an infinite number of integrating factor (resp. multiplier ). Example 6.3. Multiplier Let’s consider the differential 1-form: y 2 dx − zdy + ydz = 0 for which conditions (6.7) for this total differential equation to be exact are not valid while integrability condition (6.9) is satisfied. So this total differential equa = 1 is a tion is not exact but integrable. It may be checked that µ(X) y2 multiplier since LV µ(X) + T r (J) µ(X) = 0. 1 So, multiplied by 2 the differential 1-form may be written: y ydz − zdy z dx + = 0 ⇔ d x + =0 y2 y = φ (x, y, z) = x + The integral of this system reads: φ(X)

z =C y

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From differential equations to dynamical systems In the previous section (6.1.1) a biunivocal correspondence between a differential 1-form and a symmetrical form has been established. In dimension three since the symmetrical form consists of two differential 1-forms, any three-dimensional dynamical system may be represented by a system of two total differential equations and conversely. So, let’s consider the following system of two first order and first degree total differential equations of three variables, i.e. a system of two differential 1-form:

dφ1 = P1 (x, y, z) dx + Q1 (x, y, z) dy + R1 (x, y, z) dz = 0 dφ2 = P2 (x, y, z) dx + Q2 (x, y, z) dy + R2 (x, y, z) dz = 0

(6.10)

where Pi , Qi and Ri (i = 1, 2) are supposed to be C ∞ continuous functions in E with values in R, checking the assumptions of the Cauchy-Lipschitz = theorem (Cf. infra). The general integral 6 of this system reads: φ1 (X) φ1 (x, y, z) = C1 , φ2 (X) = φ2 (x, y, z) = C2 which define a family of space curves depending on two arbitrary constants C1 and C2 . By posing R1 P1 P1 Q1 Q1 R1 ; g (x, y, z) = ; h (x, y, z) = f (x, y, z) = Q2 R2 R2 P2 P2 Q2 Then, system (6.10) may then be transformed into a symmetrical form leading to a three-dimensional dynamical system and conversely. dx dy dz = = = dt f (x, y, z) g (x, y, z) h (x, y, z) ⇔   dX  dt 

dx dt dy dt dz dt



(6.11) 

f (x, y, z)



    g (x, y, z)  =    h (x, y, z)

6 If the integrability condition (6.7) is not satisfied for one of the two total differential equations of system (6.10) or for both, this equation is not derivable from a single integral equation equivalent to this differential equation. In such a case, the problem of determining the integral equivalents of any given total differential equation is known as the Pfaff ’s problem. See Darboux (1882), Jouanolou (1979) for details about the Pfaff’s problem.

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= φ1 (x, y, z) = C1 , General integral of dynamical system (6.11), φ1 (X) φ2 (X) = φ2 (x, y, z) = C2 , is also general integral of system (6.10) and conversely.

Necessary and sufficient condition of integrability of a 3D-dynamical system: Since the three-dimensional dynamical system (6.11) may be represented by a system (6.10) of two total differential equations its integrability depends on the integrability of each differential equation of system (6.10). Thus, if both differential equation of system (6.10) are integrable, the threedimensional dynamical system (6.11) is integrable. Remark. In the case of only one of the two differential equation of system (6.10) is integrable, the three-dimensional dynamical system (6.11) has a first integral (Cf. supra). Example 6.4. Integrable system Let’s consider the following dynamical system:  dx      f (x, y, z) x (z − y) dt dX  g (x, y, z)  =  y (x − z)   dy  = dt dt dz h (x, y, z) z (y − x) dt ⇔ dy dz dx = = = dt x (z − y) y (x − z) z (y − x) The corresponding system consisting of two differential 1-forms reads:

dφ1 = dx + dy + dz = 0 dφ2 = yzdx + xzdy + xydz = 0

Integrability conditions (6.7) are satisfied for each differential 1-form. So, each total differential equation is exact and integrable. Thus, it may be = φ1 (x, y, z) = x + y + z = C1 checked that both family of surfaces φ1 (X) = φ1 (x, y, z) = xyz = C2 represent the general integral of this and φ2 (X) system which is a family of space curves defined by their intersection.

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Remark. In the case of a three-dimensional dynamical system, the symmetric form leads to two first order and first degree differential equations involving three variables, namely x, y and z, i.e. to two differential 1-form. But according to Darboux (1878, p. 69), integration of the symmetric form and that of the following differential 1-form are “equivalent”. 6.2

First integrals – Jacobi’s last multiplier theorem

6.2.1

First integrals

Theorem 6.1. If LV φ = 0 then φ is a first integral of dynamical system (2.1). Thus, φ is constant along each trajectory curve and first integrals are drawn on the level set {φ = α} and where α is a constant. Proof.

Cf. Demazure (2000, p. 174 and next).

Theorem 6.2. A nth order differential equation, i.e. a system of n simultaneous equations of the first order, has n, and cannot have more than n, independent first integrals. Proof.

Cf. Forsyth (1885, p. 10).

Corollary 6.1. A system of n simultaneous equations of the first order is algebraically integrable if there exist (n − 1) independent algebraic first integrals. A two-dimensional dynamical system is algebraically integrable if there exists one independent algebraic first integral. Moreover, if this first integral is defined on the whole phase plane the system is said to be conservative. It will be established in part two of this book that dissipative systems may possess local first integrals. MF 14

Example 6.5. Integrable system Let’s consider the following system (Ex. 6.4):  dx  = x (z − y)   dt dy dt = y (x − z)    dz = z (y − x) dt

⇔

dy dz dx = = = dt x (z − y) y (x − z) z (y − x)

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The corresponding differential 1-form reads: x (z − y) dx + y (x − z) dy + z (y − x) dz = 0 for which conditions (6.7) for the differential 1-form to be exact and integrability condition (6.9) are satisfied. So this differential 1-form is exact and integrable. Both manifolds: = φ1 (x, y, z) = x + y + z = C1 and φ2 (X) = φ2 (x, y, z) = xyz = C2 φ1 (X) are algebraic first integrals and so the solution of this system is the curve defined by their intersection (cf. Fig 6.1).

Fig. 6.1

General integral

6.2.2

Jacobi’s last multiplier theorem

A system of n simultaneous equations of the first order is integrable via “quadratures” if there exist (n − 2) independent algebraic first integrals and a multiplier. Proof.

Cf. Jacobi (1845).

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6.3


Darboux theory of integrability

In this section Darboux theory of integrability7 of polynomial dynamical systems (Darboux, 1878) is presented with its proofs. Originally developed for three-dimensional8 homogeneous polynomial dynamical systems of degree 2 (Darboux, 1878,a,b) Darboux extended his theory to n-dimensional polynomial dynamical systems of degree m (Darboux, 1878c). 6.3.1

Algebraic particular integral – General integral

Let’s consider a polynomial dynamical system: dx2 dxn dx1 = = ··· = P1 P2 Pn

(6.12)

where Pi are algebraic polynomials of degree: m = max [deg (P1 ) , deg (P2 ) , . . . , deg (Pn )] Darboux invariance theorem (Darboux, 1878) An algebraic particular integral of a polynomial dynamical system (6.12) = 0 where φ is a C 1 in an open set U and such there is defined by φ(X) 1 and called exists a C polynomial function of degree m − 1 denoted k(X) ∈U cofactor which satisfies for all X = k(X)φ( X) LV φ(X) Proof.

Cf. Darboux (1878, p. 71 and next).

(6.13)

7 First references to Darboux theory of integrability are L. Autonne, 1891 ; H. Poincar, 1891, 1897 ; P. Painlev, 1892 ; A. R. Forsyth, Part II, Vol. II, 1900, p. 333 ; H. Dulac, 1904 ; J. Molk, 1910, p. 72 ; E. L. Ince, 1926, p. 29. 8 According to Prof. Dana Schlomiuk (Schlomiuk, 1993a, 2004, pp. 472-475) Darboux described his theory for differential equations on the projective plane. Cf. supra

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Example 6.6. Darboux invariance theorem Let’s consider the dynamical system used in Ex. 3.2: dX dt

dx

dt dy dt

=

f (x, y)

g (x, y)

=

x

−y + x2

In Ex. 3.2 it has been claimed that: ∗) = W U (X h

# x2 (x, y) ∈ R 2 y = 3

is the global unstable manifold associated with this system. Let’s prove it by application of Darboux invariance theorem. W U may be written as: = φ(x, y) = x2 − y = 0 ∗ ) = (x, y) ∈ R 2 φ(X) W U (X h 3 Thus it may be checked that: = k(X)φ( X) with LV φ(X)

= −1 k(X)

So, the unstable manifold W U is therefore globally invariant according to Darboux invariance theorem. Corollary. A general integral of polynomial dynamical system (6.12) is = 0 where φ is a C 1 in an open set U and such that for defined by φ(X) all X ∈ U =0 LV φ(X)

(6.14)

Remark. It will be pointed out in the second part that, an algebraic particular integral is an invariant manifold. That’s the reason why different names can be found in the literature such as: Darboux first integral (Schlomiuk, 1993a), Darboux polynomial (Goriely, 1999), algebraic invariant manifold (Ginoux et al., 2008) . . . The name algebraic particular integral has been introduced by Gaston Darboux (Darboux, 1878). Thus, a general integral is a first integral.

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6.3.2

General integral

Darboux theorem 1 (Darboux, 1878c) Suppose that a polynomial dynamical system (6.12) admits q algebraic 9 particular integrals ui = 0, i = 1, 2, . . . , q, with q = Mn + n − 1 where m+n−1 . Then there exists q numbers αi such that a general Mn = n q = uα1 uα2 · · · uα integral (first integral) of this system is: φ(X) q . 1 2 q = uα1 uα2 · · · uα Proof. Let’s pose φ(X) q where φ is the general integral 1 2 are algebraic particular integrals of degree hi with i = and ui = ui (X) 1, 2, . . . , q and, let’s suppose that:

i αi LV uα i (X) = αi ki (X)ui (X)

Then, = (α1 K1 + α2 K2 + . . . + αp Kp )φ(X) LV φ(X) are polynomials of degree m − 1. where Ki = ki (X) But, according to Eq. (6.14) the general integral of a dynamical system satisfies: =0 LV φ(X) So, if there exist αi numbers such that:

α1 K1 + α2 K2 + . . . + αp Kp = 0 α1 h1 + α2 h2 + . . . + αp hp = 0

⇔

 p $   αi Ki = 0  i=1 p

$   αi hi = 0 

(6.15)

i=1

System (6.15) may be satisfied if q = Mn + n − 1 algebraic particular integrals of polynomial dynamical system (6.12) are known. Then, the q = uα1 uα2 · · · uα general integral of this system has the form: φ(X) q 1 2 9 invariant

manifolds

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Remark. According to many authors (Jouanolou, 1979; Weil, 1995; Goriely, 1999; Cairo, 2000) the required number of algebraic particular integrals in order to have a general integral (first integral ) is slightly different from those proposed by Darboux (1878c). They have stated that: q = uα1 uα2 · · · uα If q > Mn a general integral of the system is: φ(X) q 1 2

This result will be used in the following examples 6.7 & 6.8. MF 15

Example 6.7. General integral Let’s consider the polynomial dynamical system (Cairo, 2000): dy dz dx = = = dt L M N where L = x(λ + Cy + z), M = y(λ + x + Az), N = z(λ + Bx + y) are non homogeneous polynomials of degree two, i.e. m = 2 and A = B = 1. According to the previous remark M3 =

m (m + 1) (m + 2) =4 6

So, this dynamical system requires q = 5 > M3 algebraic particular integrals in order to form a general integral (first integral). Thus, it may be checked that this system admits five algebraic particular integrals: ui . The plane defined by u1 = x = 0 since LV u1 = (λ + Cy + z) u1 = K1 u1 The plane defined by u2 = y = 0 since LV u2 = (λ + x + Az) u2 = K2 u2 The plane defined by u3 = z = 0 since LV u3 = (λ + Bx + y) u3 = K3 u3 The plane defined by u4 = x − Cy = 0 since LV u4 = (λ + z) u4 = K4 u4 The plane defined by u5 = y − z = 0 since LV u5 = (λ + x) u4 = K4 u5 Thus, application of Darboux Theorem 1 leads to:

α1 K1 + α2 K2 + α3 K3 + α4 K4 + α5 K5 = 0 α1 + α2 + α3 + α4 + α5 = 0

Setting α1 = −α4 = −1, α2 = 0, α3 = −α5 = C general integral reads: = φ(x, y, z) = φ(X)

u4 u1

u3 u5

C =

x − Cy x

z y−z

C

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6.3.3

Multiplier

Darboux theorem 2 (Darboux, 1878c) Suppose that polynomial dynamical system (6.12) admits p algebraic particular integrals10 ui = 0, i = 1, 2, . . . , p with p = q−1. Then, there exists p = uα1 uα2 · · · uα p numbers αi such that a multiplier of this system is µ(X) p . 1 2 p = uα1 uα2 · · · uα Proof. Let’s pose µ(X) p where µ is the multiplier and 1 2 are algebraic particular integrals of degree hi with i = 1, 2, . . . , p ui = ui (X) and suppose that:


Then, = (α1 K1 + α2 K2 + . . . + αp Kp )µ(X) LV µ(X) are polynomials of degree m − 1. where Ki = ki (X) According to Eq. (6.3) multiplier of a dynamical system satisfies: + T r (J) µ(X) =0 LV µ(X)

⇔

= −T r (J) µ(X) LV µ(X)

is a polynomial of degree −(m + 2). where µ(X) So, if there exist αi numbers such that:

α1 K1 + . . . + αp Kp = −T r (J) α1 h1 + . . . + αp hp = −m − 2

⇔

 p $   αi Ki = −T r (J)  i=1 p

$   αi hi = −m − 2 

(6.16)

i=1

Equation (6.16) may be satisfied if p = q−1 algebraic particular integrals of polynomial dynamical system (6.12) are known. Then, a multiplier of = uα1 uα2 · · · uαq−1 polynomial dynamical system (6.12) reads: µ(X) 1 2 q−1 10 invariant

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Example 6.8. Multiplier Let’s consider again the polynomial dynamical system (Cairo, 2000): dx dy dz = = = dt L M N where L = x(λ + Cy + z), M = y(λ + x + Az), N = z(λ + Bx + y) are non homogeneous polynomials of degree two, i.e. m = 2 and A = B = C = 1. According to Ex. 6.7 this dynamical system admits q = 5 algebraic particular integrals. So, in order to form a multiplier p = q −1 = 4 algebraic particular integrals are required. Thus, it may be checked that this = uα1 uα2 uα3 uα4 and only satisfy the first relation11 multiplier reads: µ(X) 1 2 3 4 of Eq. (6.16): α1 K1 + α2 K2 + α3 K3 + α4 K4 = −T r (J) Setting α1 = α2 = α3 = −1, α4 = 0 provides the multiplier : = µ(x, y, z) = µ(X)

1 1 = u1 u2 u3 xyz

Darboux theorem 3 (Darboux, 1878) Suppose that polynomial dynamical system (6.12) has a multiplier of the p = uα1 uα2 · · · uα form µ(X) p with ui = 0, i = 1, 2, . . . , p are non constant 1 2 polynomials to the power any number αi . Then, each factor equated to zero, = 0 is a algebraic particular integral (invariant manifold) of this i.e. ui (X) system. Proof.

11 Since

Cf. Darboux (1878, p. 82).

this dynamical system is not homogeneous only the first Eq. (6.16) may be satisfied. Nevertheless, Cairo and Llibre (2000) have shown that, under particular conditions, change of variables enables to transform such system into an homogeneous one

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Algebraic particular integral and fixed points

6.3.4

Darboux theorem 4 (Darboux, 1878) Suppose that polynomial dynamical system (6.12) admits p algebraic particular integrals 12 ui = 0, i = 1, 2, . . . , p with p = q − r represents manifolds which are not passing through r fixed points. Then, there exists p numbers αi such that the general integral 13 of this p = uα1 uα2 · · · uα system reads: φ(X) p . 1 2 Proof.

Cf. Darboux (1878, p. 89).

Remark. Darboux value for the bound q of algebraic particular integrals slightly differs in the literature even in Darboux publications (Darboux, 1878). This is only due to the fact that Darboux had first considered in his memoirs (Darboux, 1878) and notes in Darboux (1878a,b) homogeneous polynomial dynamical systems. Then, Darboux (1878c) generalized his theory to algebraic polynomials of degree m. So, proofs of theorems 5, 6, 7 & 8 may be deduced from the previous ones, i.e. 1,2 3 & 4 while considering that the polynomials are now homogeneous and so the number q = Mn + n − 1 of algebraic particular integrals is replaced by:

q=

Mn

+n−1

M3 6.3.5

=

where

Mn

=

m+n−2 n−1

for

n=3

m (m + 1) (m + 1)! m+1 = . = 2 2! (m − 1)! 2

Homogeneous polynomial dynamical systems of degree m

Application of these theorems to the case of three-dimensional homogeneous polynomial dynamical systems of degree m provides the following results which would be very useful in the integration of polynomial dynamical systems with flow curvature method (Cf. supra).

12 invariant 13 first

manifolds integral

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Let’s consider the homogeneous polynomial dynamical system: dy dz dx = = P Q R

(6.17)

where P , Q and R are algebraic homogeneous polynomials of degree m = max [deg (P ) , deg (Q) , deg (R)]. Then, all previous theorems may be written as follows. Darboux theorem 5 (Darboux, 1878) Suppose that polynomial dynamical system (6.17) admits q algebraic parm (m + 1) + 2. ticular integrals14 ui = 0, i = 1, 2, . . . , q, with q = 2 Then there exists q numbers αi such that the general integral15 of this sysq = uα1 uα2 · · · uα tem reads: φ(X) q . 1 2 Proof.

Cf. Darboux (1878, p. 79). α

= uα1 uα2 · · · uq q where φ is the general integral and Let’s consider φ(X) 1 2 are algebraic particular integrals of degree hi with i = 1, 2, . . . , q ui = ui (X) and, let’s suppose that: i αi LV uα i (X) = αi ki (X)ui (X)

Then,

= (α1 K1 + α2 K2 + . . . + αp Kp )φ(X) LV φ(X) are polynomials of degree m − 1. where Ki = ki (X) But, according to Eq. (6.14) the general integral of a dynamical system satisfies =0 LV φ(X)


manifolds integral

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So, if there exist αi numbers such that:

α1 K1 + α2 K2 + . . . + αp Kp = 0 α1 h1 + α2 h2 + . . . + αp hp = 0

⇔

 p $   αi Ki = 0  i=1 p

$   αi hi = 0 

(6.18)

i=1

m (m + 1) + 2 algebraic particular 2 integrals of polynomial dynamical system (6.17) are known. Then, a general q = uα1 uα2 · · · uα integral of this system has the form: φ(X) q 1 2 System (6.18) may be satisfied if

MF 16

Example 6.9. General integral of homogeneous dynamical systems Consider the polynomial dynamical system (Darboux (1878, p. 128) ; Bountis et al. (1984); Cairo (2000)): dy dz dx = = = dt L M N where L = x(Cy + z), M = y(x + Az), N = z(Bx + y) are homogeneous polynomials of degree m = 2 and A = B = C = 1. According to Th. 5: q=

m (m + 1) +2=5 2

So, this homogeneous polynomial dynamical system requires q = 5 algebraic particular integrals in order to find a general integral (first integral ). It may be checked that the planes: u1 , u2 , u3 , u4 and u5 are algebraic particular integrals for this dynamical system. The The The The The

plane plane plane plane plane

of of of of of

equation equation equation equation equation

u1 u2 u3 u4 u5

= x = 0 since LV u1 = (y + z) u1 = K1 u1 . = y = 0 since LV u2 = (x + z) u2 = K2 u2 . = z = 0 since LV u3 = (x + y) u3 = K3 u3 . = x − y = 0 since LV u4 = zu4 = K4 u4 . = y − z = 0 since LV u5 = xu5 = K5 u5 .

Thus, application of Darboux theorem 5 leads to:

α1 K1 + α2 K2 + α3 K3 + α4 K4 + α5 K5 = 0 α1 + α2 + α3 + α4 + α5 = 0

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Setting α1 = 1, α2 = 0, α3 = α4 = −1, α5 = 1 a general integral is: = φ(x, y, z) = φ(X)

u1 u3

u5 u4

x y − z = z x−y

Darboux theorem 6 (Darboux, 1878) Suppose that polynomial dynamical system (6.17) admits p algebraic m (m + 1) + 1, i.e. particular integrals 16 ui = 0, i = 1, 2, . . . , p with p = 2 p = q − 1. Then, there exists p numbers αi such that a multiplier of this p = uα1 uα2 · · · uα system reads: µ(X) p . 1 2 Proof.

Cf. Darboux (1878, p. 80).

p = uα1 uα2 · · · uα Let’s consider µ(X) where µ is the multiplier and p 1 2 ui = ui (X) are algebraic particular integrals of degree hi with i = 1, 2, . . . , p and suppose that:


Then, = (α1 K1 + α2 K2 + . . . + αp Kp )µ(X) LV µ(X) are polynomials of degree m − 1. where Ki = ki (X) According to Eq. (6.3) multiplier of a dynamical system satisfy + T r (J) µ(X) =0 LV µ(X)

⇔

= −T r (J) µ(X) LV µ(X)

So, if there exist αi numbers such that:

α1 K1 + . . . + αp Kp = −T r (J) α1 h1 + . . . + αp hp = −m − 2

⇔

 p $   αi Ki = −T r (J)  i=1 p

$   αi hi = −m − 2  i=1

16 invariant

manifolds

(6.19)

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System (6.19) may be satisfied if p = q − 1 algebraic particular integrals of polynomial dynamical system (6.17) are known. Then a multiplier of = uα1 uα2 · · · uαq−1 polynomial dynamical system (6.12) reads: µ(X) 1 2 q−1 MF 17

Example 6.10. Multiplier of homogeneous dynamical systems Let’s consider the polynomial dynamical system (Cairo, 2000): dy dz dx = = = dt L M N where L = x(x + y + z), M = y(−2x − y − z), N = z(x − 2y + z) are homogeneous polynomials of degree m = 2. According to Th. 5, i.e. in the case of homogeneous polynomial dynamical system of degree two: q = 5 = p + 1. So, this homogeneous polynomial dynamical system requires p = 4 algebraic particular integrals in order to find a multiplier. It may be checked that the planes: u1 , u2 , u3 , u4 are algebraic particular integrals for this system. The The The The

plane plane plane plane

of equation u1 = x = 0 since LV u1 = (x + y + z) u1 = K1 u1 u2 = y = 0 since LV u2 = (−2x − y − z) u2 = K2 u2 u3 = z = 0 since LV u3 = (x − 2y + z) u3 = K3 u3 u4 = 3x + 2y − 4z = 0 since LV u4 = (x − y + z) u4 = K4 u4

Thus, application of Darboux theorem 6 leads to:

α1 K1 + α2 K2 + α3 K3 + α4 K4 = −T r (J) α1 + α2 + α3 + α4 = −m − 2

which provides α1 = −α4 = −2, α2 = −1, α3 = −3. So, a multiplier is: = µ(x, y, z) = µ(X)

u24 (3x + 2y − 4z)2 = u21 u2 u33 x2 yz 3

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Darboux theorem 7 (Darboux, 1878) Suppose that polynomial dynamical system (6.17) has a multiplier of the p = uα1 uα2 · · · uα form µ(X) p with ui = 0, i = 1, 2, . . . , p are non constant 1 2 polynomials to the power any number αi . Then, each factor equated to = 0 is an algebraic particular integral 17 of this system. zero, i.e. ui (X) Proof.

Cf. Darboux (1878, p. 82).

Example 6.11. Particular integral and fixed points Let’s consider the polynomial dynamical system (Darboux, 1878, p. 90): dx dy dz = = = dt L M N where L, M , N are homogeneous polynomials of degree m = 2. Suppose that this polynomial dynamical system admits p = 2 algebraic particular integrals u1 and u2 which are not passing through r = 3 fixed points. According to Th. 5, it may be checked, in the case of homogeneous polynomial dynamical system of degree two, that:

q=

2 (2 + 1) m (m + 1) +2= +3−1=5=p+r 2 2

Thus, application of Darboux Th. 1 leads to:

α1 K1 + α2 K2 = 0 α1 h1 + α2 h2 = 0

Setting: α1 = h2 , α2 = −h1 the general integral reads: = uh2 u−h1 = C φ(X) 1 2

⇔

uh1 2 = Cuh2 1

17 invariant

manifold

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Darboux theorem 8 (Darboux, 1878) Suppose that polynomial dynamical system (6.17) admits p algebraic particular integrals 18 ui = 0, i = 1, 2, . . . , p with p = q − r represents manifolds which are not passing through r fixed points. Then, there exists p numbers αi such that a general integral 19 of this p = uα1 uα2 · · · uα system reads: φ(X) p . 1 2 Proof. 6.3.6

Cf. Darboux (1878, p. 89). Homogeneous polynomial dynamical systems of degree two

Let’s consider the homogeneous polynomial dynamical system: dx dy dz = = P Q R

(6.20)

with P , Q and R algebraic homogeneous polynomials of degree m = 2. Let’s denote by p, q, r and s the first degree polynomials, i.e. first degree invariant manifolds (planes), ui the second degree polynomials, i.e. second degree invariant manifolds (quadrics), ν the third degree polynomials, i.e. third degree invariant manifolds (cubics). Darboux theorem 9 (Darboux, 1878) (i) If pα (α = −4) is a multiplier of system (6.20) then the general integral of such system reads: ν = Cp3 with C an arbitrary constant.

(ii) If pm q m is a multiplier of system (6.20) then the general integral of such system reads: uα pβ pγ = C with C an arbitrary constant.

(iii) If pm q m rm is a multiplier of system (6.20) then the general integral of such system reads: pα q β rγ sδ = C with C an arbitrary constant.


manifolds integral

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Proof. (i) pα (α = −4) is a multiplier of system (6.20). In his proof Darboux (1878, p. 132 and next) considers that this system admits the plane z = 0 as particular solution. Let’s note that any plane of equation ax + by + cz + d = 0 could be transformed into this plane by translation and rotation of axes. Moreover, Darboux supposes20 that this plane is also a first integral . This leads to N = 0 and so system (6.20) may be written:  dx  =L   dt dy dt = M    dz = 0 dt

Since z = 0, the homogeneous polynomials L and M read L = A200 x2 + A110 xy + A020 y 2 and M = B200 x2 + B110 xy + B020 y 2 Thus, µ = z α is a multiplier21 of system (6.20) if: + T r (J) µ(X) =0 LV µ(X) But since z = 0 is first integral , i.e. z˙ = 0 this equation reduces to: T r (J) = 0

⇔

∂L ∂M + =0 ∂x ∂y

provides the following conditions for parameters Aijk and Bijk of system (6.20): A110 + 2B020 = 0

and

2A200 + B110 = 0

(6.21)

20 Although this assumption oversimplifies the demonstration it would be stated that this can be deduced from the proof (Cf. supra). 21 According to Darboux Th. 2, the degree of the multiplier is −m − 2 so, α = −4.

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110

and leads to the existence of a function ν verifying:  ∂ν    L= ∂y  ∂ν  M = − ∂x

(6.22)

By integrating Eqs. (6.22) while taking into account conditions (6.21) leads to ν = −B200

x3 y3 y2 + A200 x2 y + A110 x + A020 + K 3 2 3

Thus, it may be checked that φ = z α ν β is verifying the following equation: =0 LV φ(X) So, there exists a set of parameters for which system (6.20) admits z α ν β as a general integral22 (first integral ). The proof is stated.

(ii) pm q m is a multiplier of system (6.20). In his proof Darboux (1878, p. 132 and next) considers that this system admits the plane x = 0 and y = 0 as particular solutions. Let’s note again that any plane of equation ax + by + cz + d = 0 could be transformed into these planes by translation and rotation of axes. Moreover, Darboux supposes23 that the plane y = 0 is a first integral 24 . This leads to M = 0 and so system (6.20) may be written:  dx  =L   dt dy dt = 0    dz = N dt

22 Since Darboux considers the plane z = 0 as first integral, this integral is indeed the general integral of the system. 23 Cf. infra 24 Only one of these two planes can be considered as a first integral otherwise the system would have been completely integrated.

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Homogeneous polynomials L and M read L = A200 x2 + A110 xy + A101 xz = xA N = C200 x2 + C110 xy + C101 xz + C011 yz + C020 y 2 + C002 z 2 where A = A200 x + A110 y + A101 z = mx + ny + qz. Thus, µ = xα y β is a multiplier of system (6.20) if:

+ T r (J) µ(X) =0 LV µ(X) But since y = 0 is first integral , i.e. y˙ = 0 this equation reduces to: (α + 1)A + x

∂A ∂N + =0 ∂x ∂z

(6.23)

and provides the following conditions for parameters Aijk and Bijk of system (6.20): C101 = −(α + 2)A200 = −(α + 2)m C011 = −(α + 1)A110 = −(α + 1)n p A101 = −(α + 1) C002 = −(α + 1) 2 2

(6.24)

By integrating Eqs. (6.23) while taking into account conditions (6.24) leads to N = −(α + 1)(mxz + nyz + p

z2 ) − mxz + C200 x2 + C110 xy + C020 y 2 2

=0 Then, it may be checked that φ = xα+1 y β+1 u is verifying LV φ(X) with u = (mxz + nyz + p

C200 2 C020 2 z2 C110 )+ x + xy + y 2 α+3 α+2 α+1

So, there exists a set of parameters for which system (6.20) admits xα+1 y β+1 u as a general integral25 (first integral ). The proof is stated. 25 Cf.

infra

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(iii) pm q m rm is a multiplier of system (6.20). The proof will be emphasized by Ex. 6.12.

Example 6.12. General integral of Volterra-Lotka system Consider the polynomial dynamical system (Darboux, 1878, p. 136): dy dz dx = = = dt L M N L = x(b y+c z) = Ax, M = y(cx+b z) = By, N = z(bx+c y) = Cz are homogeneous polynomials of degree m = 2 and b, b , b , c, c , c constant coefficients. It may be checked that this polynomial system admits three algebraic particular integrals. So, a multiplier may read:

= µ(x, y, z) = pm q m rm = xα y β z γ µ(X) According to Th. 6 we have:

αA + βB + γC = −T r (J) = −(A + B + C) α + β + γ = −m − 2 = −4

This system leads to the following equation: (α + 1)A + (β + 1)B + (γ + 1)C = 0 Since polynomial A, B and C are linearly dependent the previous equation is satisfied provided that their determinant vanishes, i.e. if bb b + cc c = 0

(6.25)

According to Th. 9 and provided that condition (6.25) is satisfied a general integral of the form xα y β z γ sδ = K with s a plane and K an arbitrary constant. Let’s pose: ux + vy + wz = 0 the equation of the plane s. The Lie derivative of this plane reads: LV (ux + vy + wz) = xy(b u + cv) + yz(b v + c w) + xz(bw + c u) = 0

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So, if condition (6.25) is checked parameters u, v, w may be chosen such that:    b u + cv = 0 b v + c w = 0   bw + c u = 0 Thus, if α, β and γ verify the following set of equations compatible with Eq. (6.21):    b α + γc = 0 b β + c α = 0   bγ + cβ = 0 a general integral (first integral ) of this dynamical system reads: = φ(x, y, z) = φ(X)

xα y β z γ =K (ux + vy + wz)α+β+γ

where K is a constant and which can be also written = φ(x, y, z) = pα q β rγ sδ = K φ(X) with α + β + γ + δ = 0. Remark. Let’s note that condition (6.25) has been found again in many articles (Bountis et al., 1984; Grammaticos et al., 1990; Moulin-Ollagnier, 1997; Cairo, 2000) under the form: ABC + 1 = 0. In fact, by considering the polynomial dynamical system (Bountis et al., 1984; Cairo, 2000) called by these authors ABC system: dy dz dx = = = dt L M N where L = x(Cy + z), M = y(x + Az), N = z(Bx + y) are homogeneous polynomials of degree m = 2.

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114


DGeometry


By posing b = B, b = C, b = A, c = c = c = 1 condition (6.25) reads: ABC + 1 = 0

(6.26)

Thus, by setting A = B = 1 and C = −1 condition (6.26) is satisfied. So, according to Th. 9 this dynamical system admits a general integral of the form xα y β z γ sδ = K with K an arbitrary constant and where s is the plane of equation ux + vy + wz = 0 such that u, v and w verify:    −u + v = 0 v+w =0   w+u=0 and α, β and γ satisfying the following set of equations compatible with Eq. (6.26):    −α + γ = 0 β+α=0   γ+β =0 Posing: v = w = −u = −1, α = γ = −β = c leads to the general integral : = φ(x, y, z) = φ(X)

z(x − y − z) xy

c =K

6.3.7

Planar polynomial dynamical systems

At the end of the twentieth century Professor Dana Schlomiuk (Schlomiuk, 1993a,b) has been at the origin of a renewal of interest in the works of Gaston Darboux (Darboux, 1878). Since in his memoir Darboux described his theory of integration for differential equations on the projective plane it has been established (Jouanolou, 1979; Moulin-Ollagnier, 2002; Llibre and Zhang, 2004) that any homogeneous polynomial dynamical system in R3

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115

can be viewed as the planar projective model of a polynomial dynamical system in R2 and conversely. Then, the question of integrability of polynomial dynamical systems related to the problem of the center has been extensively studied (Schlomiuk, 2004; Dumortier et al., 2006). Let’s consider the first order and first degree differential equation (6.5): P (x, y, z)dx + Q(x, y, z)dy + R(x, y, z)dz = 0

(6.27)

P (x, y, z) = M z − N y, Q(x, y, z) = N x − Lz, R(x, y, z) = Ly − M x and where L, M and N are homogeneous polynomials in x, y, z of degree m. L(ydz − zdy) + M (zdx − xdz) + N (xdy − ydx) = 0

(6.28)

which can be written as: −L(zdy − ydz) + M (zdx − xdz) +

N (x(zdy − ydz) − y(zdx − xdz)) = 0 z

By dividing by z 2 we obtain: zdy − ydz zdx − xdz N zdy − ydz zdx − xdz +M + (x −y )=0 2 2 2 z z z z z2 x y zdx − xdz zdy − ydz Since d = = and d we have: z z2 z z2 y y x x + Md + N (xd − yd )=0 −Ld z z z z x y Replacing x by and y by leads to an equation already established z z by G. Fouret (1878): −L

(M z − N y)dx + (N x − Lz)dy = 0 Thus, by considering variable z as constant, i.e. by posing z = 1 enables to state that any homogeneous polynomial dynamical system in R3 can be considered as the planar projective model of a polynomial dynamical system in R2 . By posing p(x, y) = (M z − N y) and q(x, y) = (N x − Lz) we have: p(x, y)dx + q(x, y)dy = 0

(6.29)

Let’s note that Eq. (6.28) may also be found again from Eq. (6.29).

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116

Theorem 6.3. If φ(x, y) is first integral of (6.29) then φ xz , yz is first integral of (6.27). If φ(x, y, z) is first integral of (6.27) then φ(x, y, 1) is first integral of (6.29). Proof. According to Moulin-Ollagnier (2002) if φ(x, y, z) is first integral then φ is homogeneous of degree 0, i.e. by using Euler’s identity:

x

∂φ ∂φ ∂φ +y +z =0 ∂x ∂y ∂z

Let’s first consider that φ(x, y, z) is first integral of (6.27). So: L

∂φ ∂φ ∂φ +M +N =0 ∂x ∂y ∂z

Then, φ(x, y, z) (when z = 1) is first integral of (6.29) if: (Lz − N x)

∂φ ∂φ + (M z − N y) =0 ∂x ∂y

which can be written: z(L

∂φ ∂φ ∂φ ∂φ +M ) − N (x +y )=0 ∂x ∂y ∂x ∂y

Thus, taking into account the Euler’s identity and since it has been supposed that φ(x, y, z) is first integral of (6.27) this equation is identically zero. Now, let’s consider that φ(x, y) is first integral of (6.29). It means that: (L − N x) Then, φ

x y , z z

∂φ ∂φ + (M − N y) =0 ∂x ∂y

is first integral of (6.27) if: ∂φ L ∂φ M ∂φ + +N =0 z ∂x z ∂y ∂z

Taking into account the Euler’s identity this equation is identically zero and the proof is stated.

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MF 18

117

Example 6.13. First integral of affine and projective models According to (Cairo et al., 2003) it has been established that the corresponding affine polynomial vector field associated with the homogeneous differential system: dy dz dx = = = dt L M N

(6.30)

where L = x(Cy + z), M = y(x + Az), N = z(Bx + y) are homogeneous polynomials of degree m = 2 is: dy dx = = dt p q

(6.31)

where p = x((C − 1)y + 1 − Bx) and q = y((1 − B)x − y + A). For sake of simplicity let’s suppose that A = B = C = 1. According m(m + 1) to Darboux Th. 5, it has been stated that + 2 = 5 algebraic 2 particular integral (invariant manifolds) are necessary for obtaining a general integral (first integral ) for system (6.30). Moreover, such invariant manifolds have been determined in Ex. 6.9: = φ(x, y, z) = φ(X)

xα1 (y − z)α5 z α3 (x − y)α4

Setting α1 = α5 , α2 = 0, α3 = α4 = −α5 the general integral has the form: = φ(x, y, z) = φ(X)

x

α5

z

y−z x−y

α5 (6.32)

Thus, according to Th. 6.3 a first integral of dynamical system (6.31) may be found by posing z = 1 in Eq. (6.32). So, it may be checked that = φ(x, y) = xα5 φ(X)

y−1 x−y

α5 (6.33)

is first integral of dynamical system (6.30). Conversely, it can be shown that a first integral of dynamical system (6.30) may be found by replacing x y x by and y by in Eq. (6.33). z z

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Since any homogeneous polynomial dynamical system in R3 can be viewed as the planar projective model of a polynomial dynamical system in R2 , Th. 1, 2 and 3 may be adapted for two-dimensional polynomial dynamical systems (Cairo et al., 1999). Let’s consider the planar polynomial dynamical system: dX dt

dx

dt dy dt

=

f (x, y)

g (x, y)

(6.34)

where f and g are relatively prime algebraic polynomials of degree m = max [deg (f ) , deg (g)]. Theorem 6.4. Darboux theorem for planar dynamical systems Suppose that a polynomial dynamical system (6.34) of degree m admits q invariant algebraic curves fi with cofactors Ki , for i = 1, · · · , q. m(m + 1) λ + 1, then the function f1λ1 · · · fq q , for suitable λi ∈ R2 2 q $ not all zero, is a first integral and λi Ki = 0.

(a) If q

i=1

m(m + 1) λ + 1, then the function f1λ1 · · · fq q , for suitable λi ∈ R2 2 q $ not all zero, is a first integral and λi Ki = 0, or an integrating factor

(b) If q =

and

q $

i=1

λi Ki = −Div( ).

i=1

m(m + 1) and there exists λi ∈ R2 not all zero such that (c) If q < 2 q $ λ λi Ki = 0, then f1λ1 · · · fq q is a first integral. i=1

m(m + 1) (d) If q < and there exists λi ∈ R2 not all zero such that 2 q $ then f λ1 · · · fqλq is an integrating factor. λi Ki = −Div( ) 1 i=1

Proof.

Cf. Cairo et al. (1999).

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119

Example 6.14. Kapteyn-Bautin system Consider the following dynamical system (Kapteyn, 1911; Bautin, 1954): dX dt

dx

dt dy dt

=

f (x, y) g (x, y)

=

−y − bx2 − Cxy − dy 2

x + ax2 + Axy − ay 2

Setting parameters: a = 0, b + d = 0 and posing ∆ = C 2 + 4b(A + b) it may be checked (Schlomiuk, 1993a) that the Kapteyn-Bautin system admits three invariant algebraic curves: the straight line of equation f1 = 1 + Ay = 0 since LV f1 = (Ax) f1 = K1 f1 , the straight line of equation f2 = (C + LV f2 =

√ x ∆) − by + 1 = 0 since 2

√ ∆+C y f2 = K 2 f2 , −bx − 2

√ x ∆) − by + 1 = 0 since 2

√ ∆−C LV f3 = −bx + y f3 = K 3 f3 2

the straight line of equation f3 = (C −

√ ∆−C 1 √ and According to Th. (6.4) (b), by setting λ1 = , λ2 = A 2b ∆ √ 3 $ ∆+C √ such that λ3 = λi Ki = 0, a first integral of system (6.34) is 2b ∆ i=1 1

φ(x, y) = (1+Ay) A

√ C+ ∆ x − by + 1 2

√

∆−C √ 2b ∆

√ C− ∆ x − by + 1 2

√ ∆+C √ 2b ∆

Remark. In the second part of this book it will be stated that such linear invariant curves may be directly found by using the Flow Curvature Method.

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PART 2


It has adopted the geometry most advantageous to the species or, in other words, the most convenient. — H. Poincaré —

121

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122

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Chapter 7


“Geometry is not true, it is advantageous.” — H. Poincaré —

In the second part of this book a new approach for studying ndimensional dynamical systems called flow curvature method is presented. Based on the use of Differential Geometry1 it consists in considering trajectory curves, integral of such systems as curves in Euclidean n-space which possess local metrics properties of curvatures enabling to define a manifold called: flow curvature manifold. Then, it will be established that all the features of n-dimensional dynamical systems, presented in the first part of this work, such as fixed points stability, local bifurcations, center manifolds, normal forms, invariant manifolds and integrability may be deduced from this flow curvature manifold which contains information about the dynamics of such systems. Let’s consider a set of differential equations defined in a compact E = [x1 , x2 , ..., xn ]t ∈ E ⊂ Rn : included in R with X dX X) = ( (7.1) dt

t X) = f1 (X), f2 (X), ..., fn (X) where ( ∈ E ⊂ Rn defines in E a velocity vector field whose components fi , supposed to be C ∞ continuous functions in E with values in R, are checking the assumptions of the Cauchy-Lipshitz theorem (Cf. infra) 1 For

a History of Differential Geometry see Struik (1933a,b) and Coolidge (1947). 123

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124

7.1


DGeometry


Concept of curves – Kinematics vector functions

A presentation of these concepts and definitions may be found, for example, in Forsyth (1912), Struik (1961), Kreyszug (1959), Delachet (1964), Gluck (1966, 1967), Stoker (1969), Lipshutz (1969), Thorpe (1979), Postnikov (1981) or Gray et al. (2006). 7.1.1

Trajectory curve

According to Poincaré (1886, p. 168), the curve solution to dynamical system (6.1) “will define the motion of a variable point in a space of dimension n” and thus, would be associated with the coordinates, i.e. with the position, of a point M at the instant t. Such integral curve defined by the (t) of the scalar variable t represents the trajectory of M. vector function X Definition 7.1. A smooth parametrized 2 curve in Rn is a smooth map (t) : [a, b] → Rn from a closed interval [a, b] into Rn . A map is said to be X smooth or infinitely many times differentiable if the coordinate functions = [x1 , x2 , . . . , xn ]t have continuous partial derivatives x1 , x2 , . . . , xn of X of any order. (t) integral of n-dimensional dynamRemark. Thus, trajectory curves X ical systems (7.1) satisfying the assumptions of the Cauchy-Lipschitz theorem may be regarded as n-dimensional smooth curves, i.e. smooth curves in Euclidean n−space time parametrized. 7.1.2

Instantaneous velocity vector

(t) is the vector function V (t) The time derivative of the trajectory curve X of the scalar variable t, i.e. the instantaneous velocity vector of the mobile M at the instant t X) (t) = dX = ( V dt

(7.2)

The instantaneous velocity vector V (t) is supported by the tangent to the trajectory in any point but the fixed points. 2 with

any kind of parametrization

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125

Instantaneous acceleration vector

(t) is the vector funcTime derivative of the instantaneous velocity vector V tion γ (t) of the scalar variable t, i.e. the instantaneous acceleration vector of mobile M at instant t γ (t) =

dV dt

(7.3)

Since functions fi are supposed to be C ∞ functions in a compact E included in Rn , it is possible to calculate the total derivative of velocity vector field V (t) defined by (7.1). While using the chain rule, the Fréchet derivative reads: ∂X ∂ dV ∂X = = DX ∂t dt ∂t ∂X

(7.4)

is the functional Jacobian matrix J of system (7.1), it follows Since DX from Eqs. (7.3) and (7.4) that: γ = J V

7.2

(7.5)

Gram-Schmidt process – Generalized Fr´ enet moving frame

There are many moving frames along a trajectory curve but among them Frénet (1852) frame is related to local metrics properties of curvatures. In this sub-section generalized Frénet frame for n-dimensional trajectory curves in Euclidean n-space is recalled. Let’s suppose that the trajectory (t), parametrized in terms of time, is of general type in Rn , i.e. curve X (n−1)

¨ (t), . . . , X ˙ (t), X (t), are linearly that the first n − 1 time derivatives: X (t) of independent for all t. A moving frame along a trajectory curve X n general type in R is a collection of n vectors u1 (t), u2 (t), . . . , un (t) along (t) forming an orthogonal basis, such that: X ui (t) · uj (t) = 0

(7.6)

for all t and for i = j. These vectors un (t) may be determined according to the Gram-Schmidt orthogonalization process described below.

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7.2.1

Gram-Schmidt process (n−1)

¨ (t), . . . , X ˙ (t), X (t) be linearly independent vectors for all t in Let X n R . According to Gram-Schmidt process (Lichnerowicz (1950, p. 30), Gluck (1966)) the vectors u1 (t), u2 (t), . . . , un (t) forming an orthogonal basis are defined by: ˙ (t) u1 (t) = X

¨ (t) u1 (t) · X u1 (t) · u1 (t)  

¨ (t) − u2 (t) = X

u1 (t)

  ... ... ... (t) (t) u X X u (t) · (t) · 1 2 (t) −   u1 (t) −   u2 (t) u3 (t) = X u1 (t) · u1 (t) u2 (t) · u2 (t) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . .  (n)

(t) − un (t) = X

n−1 i=1

7.2.2

(n)



(t)   ui (t) · X  ui (t)  ui (t) · ui (t)

(7.7)

Generalized Fr´ enet moving frame

Starting from the vectors u1 (t), u2 (t), . . . , un (t) forming an orthogonal (t) of basis, generalized Frénet moving frame for the trajectory curve X n general type in R may be built. Thus derivation with respect to time t leads to the generalized Frénet formulas in Euclidean n-space: u˙ i (t) = v

n

αij uj (t)

j=1

+ + + + + + + ˙ + with j = 1, 2, . . . , n and where v = +X + represents the Euclidean + = +V norm of the velocity vector field. Moreover, according to Eq. (7.6) it comes that: u˙ i (t) · uj (t) + ui (t) · u˙ j (t) = 0

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So, αii = 0 and αij = 0 for j < i − 1. Thus, only αi,i+1 = −αi+1,i are not identically zero. Let’s pose: κ1 = α12 ,

κ2 = α23 ,

...,

κn−1 = αn−1,n

The generalized Frénet formulas associated with a trajectory curve in Euclidean n-space read:  u˙ 1 (t) = vκ1 u2 (t)      u˙ 2 (t) = v [−κ1 u1 (t) + κ2 u3 (t)]   ˙ u3 (t) = −vκ2 u2 (t) ·····················     u˙ n−1 (t) = v [−κn−2 un−2 (t) + κn−1 un (t)]   ˙ un (t) = −vκn−1 un−1 (t)

(7.8)

Functions κ1 , κ2 , . . . , κn−1 are called curvatures of trajectory curve (t) of general type in Rn and κn−1 is analogous to the torsion (Cf. supra). X 7.3

Curvatures of trajectory curves – Osculating planes

Within the framework of Differential Geometry, n-dimensional smooth curves, i.e. smooth curves in Euclidean n−space are generally defined by a regular parametric representation in terms of arc length also called natural representation or unit speed parametrization. According to Gluck (1966, 1967) local metrics properties of curvatures may be directly deduced from curves parametrized in terms of time and so natural representation is not necessary. Theorem 7.1 (Curvatures). (t) integral of n-dimensional dyThe curvatures of trajectory curves X namical system (7.1) read: κn−1 =

un (t) = n−1 , u1 (t) un−1 (t) i=1

(n)

¨ . . . , X ˙ X, X,

(7.9) u1 ui un−1

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128

Proof. By considering the basis vectors u1 (t), u2 (t), . . . , uk (t) with 1 ≤ k ≤ n − 1, defined according to Gram-Schmidt process (7.7) and while using identity (A.10) established in appendix:

(n)

˙ X, ¨ . . . , X X,

(n)

¨ . . . , X ˙ X, where X,

= u1 u2 . . . un

(n) ... ¨ ˙ ∧ ...∧ X ∧X represents the inner prod= X· X

uct, the proof is stated.

Corollary 7.1. A n-dimensional trajectory curve has (n − 1) curvatures. Theorem 7.2 (Osculating plane). (t) : [a, b] → Rn , the trajectory curve integral of n-dimensional Let X dynamical system (7.1) be a smooth time parametrized curve of general type (n−1)

¨ (t), . . . , X ˙ (t) , X (t), in Rn , such that its first n − 1 time derivatives: X 3 to are linearly independent for all t. Then, the osculating plane Π(X) ∗ the trajectory curve X (t) is the plane passing through a fixed point X (n−1)

(t) the equation of which ˙ (t) , X ¨ (t), . . . , X and spanned by the vectors X reads: −X ∗ · = X Π(X)

(n−1)

(n−1)

− X , X, ˙ X, ¨ . . . , X = X ∗

Proof.

Proof is left as an exercise

˙ ∧ X ¨ ∧ . . . ∧ X X =0

(7.10)

Corollary 7.2. A (n − 1)-dimensional osculating plane passes through n points. 3 The

word “osculate” which comes from Latin “osculari” – to kiss has been introduced by Tinseau (1780).

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7.4

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129

Curvatures and osculating plane of space curves

(t) : [a, b] → Rn be the space curve integral of a three-dimensional Let X dynamical system be a smooth time parametrized curve of general ...type in ˙ ¨ n (t) are (t) and X R , i.e. such that the first time derivatives: X (t), X linearly independent for all t.

7.4.1

Fr´ enet trihedron – Serret-Fr´ enet formulae

By normalizing the basis vectors u1 (t), u2 (t) and u3 (t) obtained with the Gram-Schmidt process, the so-called Frénet trihedron for space curves may be deduced. Hence, it may be stated that: u1 (t) u2 (t) u3 (t) , , = (τ , n, b) u1 (t) u2 (t) u3 (t) where τ , n and b are the tangent, normal and binormal unit vectors respec˙ (t), X ¨ (t) and tively. Let’s notice that the three first time derivatives: X ... (t) represent the velocity, acceleration and over-acceleration vector fields X (t), γ (t) and γ˙ (t) respectively. namely: V (t) and diThus, the Frénet (1852) frame built from the space curve X rected towards the motion of mobile M consists in a unit tangent vector τ to the space curve in M, a unit normal vector n, i.e. the principal normal in M directed towards the interior of the concavity of the space curve and to the space curve in M so that the trihedron a unit binormal vector β (τ , n, β) is direct since the instantaneous velocity vector V is tangent to (t). any point M to the space curve X The unit tangent vector is defined as: ˙ V X +=+ + τ = + + + + ˙ + +V + +X +

(7.11)

The unit binormal vector is defined, as: ¨ ˙ = +γ ∧ V + = +X ∧ X + β + + ¨ ˙ + + +γ ∧ V + +X ∧ X +

(7.12)

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and the unit normal vector, as: ∧ τ n = β

Fig. 7.1

(7.13)

Osculating plane

According to (7.8) the first derivatives of the unit vectors τ , n and β can be represented as a linear combination of these vectors, i.e. the so-called called Serret-Frénet formulae:  d τ = νκ1 n    dt

 d n = ν −κ τ + κ β (7.14) 1 2 dt     dβ n dt = −νκ2 + + + + + ˙ + + + where ν = +X + = +V + represents the Euclidian norm of the instantaneous velocity vector. From Th. 7.2, osculating plane definition may be found again.

7.4.2

Osculating plane

The osculating plane to the space curve integral of a three-dimensional dy ∗ and spanned namical system is the plane passing through a fixed point X by the instantaneous velocity and acceleration vectors the equation of which reads:

¨ = 0 ¨ = X −X ∗ , X, ˙ X ˙ ∧ X = X −X ∗ · X Π(X)

(7.15)

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Remark. The osculating plane is the plane parallel to both unit tangent and normal vectors to a space curve passing through a fixed point. From the generalized Frénet formulas (7.7) and expression (7.9) for curvatures, the first and second curvatures of space curves, i.e. curvature and torsion may be found again. 7.4.3

Curvatures of space curves

For low dimensions two and three the concept of curvatures may be simply exemplified. A three-dimensional4 curve, i.e. a space curve has two curvatures: curvature and torsion which are also known as first and second curvature. Curvature measures, so to speak, the deviation of the curve from a straight line in the neighborhood of any of its points. While the torsion measures, roughly speaking, the magnitude and sense of deviation of the curve from the osculating plane in the neighborhood of the corresponding point of the curve, or, in other words, the rate of change of the osculating plane. Physically, a three-dimensional curve may be obtained from a straight line by bending (curvature) and twisting (torsion). Theorem 7.3 (first curvature). The (first) curvature κ1 of the space5 curve integral of the threedimensional dynamical system (7.1) is given by:

˙ X ¨ X,

κ1 =

u2 = u1 u1 u1 3

+ + + + + ˙ + ¨ + + +X ∧ X + +γ ∧ V + = + +3 = + +3 + + + ˙ + +V + +X +

(7.16)

Proof. While replacing basis vectors u1 (t) and u2 (t) resulting from the Gram-Schmidt process (7.7) and by using the Lagrange identity: +2 + + ˙ 2 2 ¨ + ∧X u1 u2 = +X + the proof is stated.

4A

two-dimensional curve, i.e. a plane curve has a torsion vanishing identically. curvature of a two-dimensional curve, i.e. of a plane curve is identically defined.

5 First

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Corollary 7.3. A necessary and sufficient condition that the curvature is identically zero is that the curve be a straight line. Proof.

Cf. Graustein (1935, pp. 27, 32, 36 and next).

Theorem 7.4 (second curvature). The second curvature, i.e. torsion κ2 of the space curve integral of the three-dimensional dynamical system (7.1) is given by: ... ˙ ¨ X, X, X

... ¨ ˙ X · X ∧X γ˙ · γ ∧ V u3 = = = − κ2 = + + + + 2 2 2 2 u1 u2 + + ˙ u1 u2 + ¨ + + +γ ∧ V +X ∧ X +

Proof.

Using identity (A.10):

(7.17)

... = u1 u2 u3 , the Gram˙ X, ¨ X X,

Schmidt orthogonalization process (7.7) for the expression of vectors u1 (t) +2 + + ˙ 2 2 ¨ + ∧X and u2 (t) and the Lagrange identity u1 u2 = +X + enables to state the proof. Corollary 7.4. A necessary and sufficient condition that the torsion is identically zero is that the curve be a plane curve. Proof.

Cf. Graustein (1966) pp. 27, 32, 36 and next.

Remark. A trajectory curve for which the torsion is identically zero has its osculating plane stationary. In this case, the trajectory curve is a plane curve. The name torsion was first used by L.L. de la Vallée (1825). Curvature and torsion are also known as first and second curvature, respectively and a twisted curve (space curve) is therefore called a curve of double curvature.

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7.5

133

Flow curvature method

Flow curvature method consists in using the highest curvature of the flow, i.e. the (n − 1)th curvature of trajectory curve integral of n-dimensional dynamical system (7.1) for defining a manifold associated with this system. 7.5.1

Flow curvature manifold

Definition 7.2. The location of the points where the (n− 1)th curvature of the flow, i.e. the curvature of the trajectory curve integral of n-dimensional dynamical system (7.1) vanishes defines a (n−1)-dimensional flow curvature manifold the equation of which reads: =X ˙ · φ(X)

(n) ... ∧ ...∧ X ¨ ∧ X X

(n) ... ..., X ˙ X, ¨ X, = det X,

=0

(7.18)

(n)

represents the time derivatives of X = [x1 , x2 , ..., xn ]t . where X Remark. According to Corollary 7.1 the (n − 1)th curvature of the flow corresponds to the highest curvature. In what follows flow curvature manifold will always refer to the highest curvature of the flow. 7.5.2

Flow curvature method

Since (n − 1)th curvature of the trajectory curve integral of n-dimensional dynamical system (7.1) only involves the instantaneous velocity vector field ˙ (t) and its time derivatives, flow curvature method characterizes local X metrics properties of a trajectory curve the equation of which is generally analytically unknown. It will be established in the subsequent part of this book that main features of n-dimensional dynamical systems, presented in the first part, such as: • fixed points and their stability, • local bifurcations of codimension one, • center manifold equation, • normal forms, • linear invariant manifolds (straight lines, planes, hyperplanes).

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may be straightforwardly deduced from this flow curvature manifold which contains information about the dynamics of such systems. Moreover, in the case of singularly perturbed systems or slow-fast dynamical systems it will also be established that the flow curvature manifold directly provides the slow invariant manifold analytical equation associated with such systems. Then, starting from the flow curvature manifold, it will be demonstrated how to find again the corresponding dynamical system, solving thus the inverse problem.

DGeometry

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Chapter 8

Dynamical Systems

“. . . it may happen that small differences in the initial conditions produce very great ones in the final phenomena . . . ” — H. Poincaré —

In this chapter it will be demonstrated that stability theorems of lowdimensional two and three dynamical systems presented in the first part of this book (Ch. 2) may be deduced from the flow curvature manifold. 8.1

Phase portraits of dynamical systems

Thus, it will be established in this section that fixed points as well as their stability may be deduced from the flow curvature manifold. Let’s consider a system of differential equations defined in a compact E = [x1 , x2 , ..., xn ]t ∈ E ⊂ Rn : included in R with X dX X) = ( (8.1) dt

t X) = f1 (X), f2 (X), ..., fn (X) where ( ∈ E ⊂ Rn defines in E a velocity vector field whose components fi , supposed to be C ∞ continuous functions in E with values in R, are checking the assumptions of the Cauchy-Lipshitz theorem (Cf. infra) 8.1.1

Fixed points

∗ of n-dimensional dynamical system (8.1) Theorem 8.1. Fixed points X ∗ ) = 0. are singular solutions of the flow curvature manifold, i.e. φ(X 135

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136

Proof. According to Def. 2.2, the velocity vector field evaluated at the ∗ reads: fixed points X X = ( ∗ ) = 0 V ∗ So, the flow curvature manifold (7.18) evaluated at the fixed points X vanishes identically: ˙ · φ(X ) = X ∗

(n) ... ∧ ...∧ X ¨ ∧ X X

∗

X )· = (

(n) ... ∧ ...∧ X ¨ ∧ X X

=0

∗ of the flow curvature manifold are the Corollary 8.1. Fixed points X points defined by the set of equations:

∗) = 0 φ(X ∗) = 0 ∇φ(X

(8.2)

Thus fixed points may be deduced from the flow curvature manifold which can be conversely considered as a manifold of fixed points. Example 8.1. Flow curvature manifold and fixed points = y 2 − x (x − a)2 = 0 with a > 0 Consider the manifold (curve): φ(X) (Piskounov, 1966, p. 338). According to Th. 8.1 the fixed points coordinates are singular solutions to system (8.1) whose second equation reads: ∂φ X ∗) = 0 ∇φ(

⇔

∂x ∂φ ∂y

= (x − a) (a − 3x) = 0 = 2y = 0

This reduced system leads to two fixed points: I (a, 0) and J (a/3, 0). Then, by using the first equation of (8.2), it is obvious that the former is a fixed point of the manifold, while the latter is not since φ (a/3, 0) = 3 −4 (a/3) = 0.

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8.1.2

137

Stability theorems

Theorem 8.2. ∗ the flow curvature manifold of nIn the vicinity of the fixed points X dimensional dynamical system (8.1) merges with its Lie derivative and with its osculating plane. Proof. Let’s consider the flow curvature manifold (7.18) associated with the flow of the n-dimensional dynamical system (8.1): ˙ · =X φ(X)

(n) ... ∧ ... ∧ X ¨ ∧ X X

(n) ... ..., X ¨ X, ˙ X, = det X,

=0

˙ X). = L φ(X) = dφ = φ( Its Lie derivative1 reads: ψ(X) So, while V dt using determinant derivative theorem we have: = L φ(X) =X ˙ · ψ(X) V

(n+1) ... ∧ ...∧ X ¨ ∧ X X

(n+1) ... ..., X ˙ X, ¨ X, = det X,

=0

(8.3)

∗ the functional jacobian Moreover, in the vicinity of the fixed points X dJ = 0. matrix may be considered as locally stationary: dt (n+1)

˙ where J n represents the nth power of J. As an example, = J nX So, X ¨ ˙ = JX ⇔ γ = J V . Then, it follows that X (n+1)

(n)

= JJ n−1 X ˙ = J X X

(8.4)

(n+1)

Replacing X

in expression (8.3) leads to:

= L φ(X) =X ˙ · ψ(X) V 1 See

(n) ... ∧... ∧ J X ¨ ∧ X X


(n) ... ...,J X ˙ X, ¨ X, = X,

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Then, identity established (A.16) in appendix leads to: ˙ · = L φ(X) = T r (J) X ψ(X) V

(n) ... ∧ ...∧ X ¨ ∧ X X

(8.5) = T r (J) φ(X)

where T r (J) represents the trace of the functional jacobian matrix. Thus, ∗ , the flow curvature manifold and its in the vicinity of the fixed point X Lie derivative are merged. ∗ is Now, let’s suppose without loss of generality that the fixed point X ∗ = 0. According to Eq. (2.2) the velocity vector field at the origin, i.e. X X) in the vicinity of X ∗ reads: ( X) = J(X ∗ )X = JX V = ( by (8.4) in the flow curvature So, replacing all the time derivatives of X manifold leads to: = JX · φ(X)

(n−1)

˙ ∧ J X ¨ ∧ . . . ∧ J X JX

=0

Then, while using identity (A.15) established in appendix and the osculating plane equation (7.10) passing through the origin, the flow curvature manifold reads: = |J| X · φ(X)

(n−1)

¨ ∧ . . . ∧ X ˙ ∧ X X

= Det (J) Π(X)

(8.6)

∗ , flow curvature manifold and Thus, in the vicinity of the fixed point X osculating plane are merged. The proof is stated. Corollary 8.2. Trace and determinant of the functional jacobian matrix associated with a n-dimensional dynamical system (8.1) are defined by: φ( X) L

V = T r (J) φ(X) LV φ(X) ⇔ p = T r (J) = φ(X)

∗ X φ(X) = Det (J) Π(X) φ(X) ⇔ q = Det (J) = ∗ Π(X)

X

(8.7)

(8.8)

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139

∗ are Remark. According to Th. 8.1 and Corollary 8.2 fixed points X singular solutions of both flow curvature manifold and its Lie derivative. Moreover, osculating planes are passing through the fixed points (Th. 7.2). So, Eqs. (8.7) & (8.8) lead to indeterminate forms which may be evaluated, ∗ + ε and where ε → 0. ∗ with X in practice, by replacing X Thus, these results may be used to state the fixed points stability of dynamical systems of dimension two or three. But before, it will be established below that the discriminant of the functional jacobian matrix associated with such systems may be deduced from the flow curvature manifold which contains much information about the dynamics of the system. Theorem 8.3. The Hessian of the 1st flow curvature manifold associated with a two-dimensional dynamical system enables to discriminate focus from saddles (resp. nodes). ∗ is a If Hφ(X ∗ ) ≤ 0 then both eigenvalues are real and the fixed point X saddle or a node. If Hφ(X ∗ ) > 0 then both eigenvalues are complex conjugated and the ∗ is a focus. fixed point X Proof. According to Th. 7.3 and Def. 7.2 the 1st flow curvature manifold of a two-dimensional dynamical system reads: + + + + + ˙ + ¨ + + +X ∧ X +γ ∧ V + + κ1 = + +3 = + +3 + + + ˙ + +V + +X +

⇔

+ + =+ + φ(X) +γ ∧ V +=0

Let’s focus on extrema of the 1st flow curvature manifold which may be denoted H . obtained while using the Hessian2 of φ(X) φ(X ) 2 ∂ φ ∂x2 Hφ(X ) = 2 ∂ φ ∂y∂x

2 See

Appendix for definition of Hessian.

∂2φ ∂x∂y ∂2φ ∂y 2

(8.9)

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= +γ ∧ V + . According to Eq. (7.12) the unit binormal vector reads: β + + + +γ ∧ V From the scalar product · γ ∧ V β + + β·β = + + + +γ ∧ V

=1

it can be deduced that + + + + + = β · γ ∧ V +γ ∧ V and thus the 1st flow curvature manifold may be written: + + + =+ φ(X) +γ ∧ V + = β · γ ∧ V

=0

∗ is at the Let’s suppose without loss of generality that the fixed point X ∗ X) origin, i.e. X = 0. According to Eq. (2.2) the velocity vector field ( ∗ ∗ X) = ( = J(X )X = JX and according reads: V in the vicinity of X . So, while using the to Eq. (7.5) the acceleration vector field: γ = J V identity (A.15) established in appendix and the osculating plane equation (7.10) passing through the fixed point, i.e. the origin, the 1st flow curvature manifold reads: · γ ∧ V = β · JV =β ∧ JX = |J| β · V ∧X φ(X) · V ∧X = qΠ(X) =0 = qβ

(8.10)

∗ reads: Thus the Hessian matrix elements evaluated at the fixed point X

∂ 2 φ ∂X ∂g ∂V ∧ = −2q ; = 2q β · 2 ∂x X ∗ ∂x ∂x ∂x

∂ 2 φ ∂X ∂f ∂V ∧ = +2q ; = 2q β · 2 ∂y X ∗ ∂y ∂y ∂y

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141

∂ X ∂ V ∂ X ∂f ∂g ∂ 2 φ ∂ V · ∧ + ∧ = − = β q; ∂x∂y X ∗ ∂x ∂y ∂y ∂x ∂x ∂y Then, replacing into the Hessian leads to: 2 ∂ φ ∂x2 Hφ(X ) = 2 ∂ φ ∂y∂x

∂2φ ∂x∂y ∂2φ ∂y 2

= −q 2 ∆

(8.11)

Remark. This approach which enables to state the fixed points stability of two-dimensional dynamical systems3 corresponds to the Poincaré topographic system introduced in his memoirs entitled Sur les courbes définies par une équation différentielle, Poincaré (1881, p. 384). MF 20

Example 8.2. Duffing oscillator Let’s consider the Unforced Duffing Oscillator V

x˙ y˙

=

f (x, y) g (x, y)

=

y

x − x3

This system has three fixed points, the origin which is a saddle and two centres: I± (±1, 0) According to Th. 7.3 and Def. 7.2 the 1st flow curvature manifold reads: = y 2 − x2 φ(X)

2 x2 − 1 + 3y 2 = 0

Thus, the fixed points coordinates may be found again (Cf. Th. 8.1) while solving the whole4 system (8.2). The Hessian evaluated at the origin reads: Hφ(X ∗ ) = −4. So, Th. 8.3 enables to state that the origin is a saddle or a node. According to Th. 7.2 the osculating plane equation is: = y 2 + x4 − x2 = 0 Π(X) 3 See

§ 2.6. being only solutions of the second equation of system (8.2) may be not fixed points. - √ In this example, solving second equation of (8.2) provides two new points 3, 0 which are not solutions of the first equation and thus they are not fixed ±1 points of the Duffing oscillator. Cf. infra Ex. 8.1. 4 Points

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Then the determinant of the functional jacobian matrix associated with ∗ (0, 0) + ∗ (0, 0) by X this system evaluated at the origin on replacing X ε (ε, ε) and where ε → 0 reads: φ(X) q = Det (J) = ∗ Π(X)

= −1 − ε2 −→ −1 ε→0

ε X +

Thus, as pointed out in Fig. (8.1) the origin is a saddle point.

Y 2

1

0

X

1

2 2

1

Fig. 8.1

0

1

2

Duffing oscillator

Theorem 8.4. The relative Hessian of the 2nd flow curvature manifold enables to discriminate saddle-nodes from saddle-foci. ∗ is a ¯ ∗ ≥ 0 then both eigenvalues are real and the fixed point X If H φ(X ) saddle-node. ¯ ∗ < 0 then two eigenvalues are complex conjugated and the fixed If H φ(X ) point X ∗ is a saddle-focus or a center.

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143

Proof. According to Th. 7.4 and Def. 7.2 the 2nd flow curvature manifold of a three-dimensional dynamical system reads: ... ¨ ˙ X · X ∧X γ˙ · γ ∧ V κ2 = + +2 +2 = − + + + ˙ + ¨ + + +γ ∧ V + +X ∧ X

⇔

= γ˙ · γ ∧ V φ(X)

=0

Let’s focus on the extrema of the 2nd flow curvature manifold which denoted H . may be obtained while using the Hessian5 of φ(X) φ(X) 2 ∂ φ ∂x2 2 ∂ φ Hφ(X ) = ∂y∂x ∂2 φ ∂z∂x

∂2 φ ∂2φ ∂x∂y ∂x∂z ∂2 φ ∂y 2

∂2φ ∂y∂z

∂2φ ∂2φ ∂z∂y ∂z 2

(8.12)

The Hessian of the 2nd flow curvature manifold evaluated at the fixed ∗ ) where p = T r (J), q = Det (J) ∗ reads: H ∗ = −2q 2 Rφ(X point X φ(X ) are the trace and determinant of the functional Jacobian matrix and R = 4P 3 + 27Q2 is the discriminant of the characteristic equation. But, ∗ ) = 0. ∗ ) = 0, then H ∗ = −2q 2 Rφ(X since according to Th. 8.1 φ(X φ( X ) So, in order to state the nature of the fixed points of a three-dimensional dynamical system the Hessian has to be divided by the 2nd flow curvature manifold.

¯ = Let’s pose: H φ(X)

Hφ(X)

φ(X)

and call relative Hessian this ratio.

According to Def. 7.2, the 2nd flow curvature manifold reads: = γ˙ · γ ∧ V φ(X)

=0

∗ is at the Let suppose without loss of generality that the fixed point X ∗ origin. Thus, in the vicinity of X the velocity vector field reads according X) = J X, the acceleration vector field = ( = J(X ∗ )X to Eq. (2.2): V reads γ = J V according to Eq. (7.5) and the over-acceleration vector 5 See

appendix for definition of Hessian.

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field reads γ˙ = Jγ according to (8.4). So, while using the identity (A.15) established in appendix and the osculating plane equation (7.10) passing through the fixed point, i.e. the origin, 2nd flow curvature manifold reads: ∧X = γ˙ · γ ∧ V = Jγ · J V ∧ JX = |J| γ · V φ(X) ∧X = qΠ(X) =0 = qγ · V

(8.13)

Then, by using the same approach as previously it can be proved that: ¯ ∗ = −2q 2 R H φ(X )

(8.14) 6

Remark. For dimension greater than five , discriminant of higherdimensional dynamical systems can not be analytically determined according to Galois Theory. In this case, only numerical computations enable to state the fixed points stability. MF 21

Example 8.3. Lorenz model (Lorenz, 1963) Let’s consider the Lorenz model (Cf. Ex. 2.11.):       x˙ f (x, y, z) σ(y − x)  g (x, y, z)  =  −xz + rx − y  V  y˙  = z˙ h (x, y, z) xy − bz with the classical set of parameters σ = 10, b = 8/3, r = 28 this dynamical system exhibits three unstable fixed points: O(0, 0, 0) and I± (±

' ' b(r − 1), ± b(r − 1), r − 1)

the stability of which is well-known (Guckenheimer and Holmes, 1983, p. 94). The origin O is a saddle-node and both I± are saddle-foci. Evaluated at the origin the relative Hessian is positive while it is negative for both fixed points I± . So, Th. 8.4 enables to confirm the stability of each fixed point. Moreover, it may be stated, according to Corollary 8.2 that the real eigenvalue associated with the fixed points I± is negative. 6 Four degree algebraic equations have no more discriminant (excepted bi-squared equation).

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Chapter 9

Invariant Sets

“Invention consists in avoiding the constructing of useless contraptions and in constructing the useful combinations which are in infinite minority.” — H. Poincaré —

As pointed out in the first part (Ch. 3) invariant manifolds play a very important role in stability and structure of dynamical systems and especially for singularly perturbed systems. It will be established in this chapter that flow curvature manifold enables to find out linear invariant manifolds1 (straight lines, planes, hyperplanes), i.e. first degree algebraic invariant manifolds while extatic algebraic manifolds provide higher degree algebraic invariant manifolds. Thus, using Darboux theory of integrability, such manifolds invariant with respect to the flow of n-dimensional dynamical systems will be used in order to built the corresponding first integrals.

9.1

Invariant manifolds

Let’s focus on two concepts introduced by Gaston Darboux in his memoir (1878 page 71 and next) with the meaning of algebraic particular integral (Darboux invariance theorem, Ch. 6), general integral (Corollary, Ch. 6). Theorem 9.1. An algebraic particular integral in the sense of Darboux is an invariant manifold. = 0 be a manifold where φ is a C 1 in an open set U Proof. Let φ(X) and such there exists a C 1 polynomial function of degree m − 1 denoted = k(X)φ( called cofactor satisfying for all X ∈ U : L φ(X) X). k(X) V 1 This

terminology has been suggested by Professor Dana Schlomiuk. 145

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The Lie derivative of the vector field V is also called derivative along the X). Thus ∀ X ∈ φ, Φt (X) ∈ φ for flow Φt generated by the vector field ( all t ∈ R. According to Def. 3.1 manifold φ is therefore an invariant set. Theorem 9.2. A general integral in the sense of Darboux is a first integral. Proof. An algebraic general integral of a dynamical system is defined by ∈ U: = 0 where φ is a C 1 in an open set U and such that for all X φ(X) = 0. According to Th. 6.1 this is the definition of a first integral. L φ(X) V

9.1.1

Global invariance

= 0 for a flow Φt : D → Definition 9.1. A global invariant manifold φ(X) Rn associated with the dynamical system (8.1) is such that: = k(X)φ( X) LV φ(X)

∈ φ for all ∀X

t∈R

(9.1)

Remark. A first integral is globally invariant (Cf. Th. 9.2 and Def. 9.1). Example 9.1. Global invariance According to Darboux invariance theorem it has been established in Ex. 6.6 that the dynamical system of Ex. 3.2 (Cf. infra): dX dt

dx

dt dy dt

= φ(x, y) = admits φ(X)

=

f (x, y) g (x, y)

=

x

−y + x2

x3 − y as global invariant manifold. 3

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9.1.2

147

Local invariance

= 0 for a flow Φt : D → Rn associated A local invariant manifold φ(X) with the dynamical system (8.1) is such that: = k(X)φ( LV φ(X) X)

∈ φ for ∀X

|t| < T with T > 0

(9.2)

Remark. Besides this local invariance definition which is time-dependant a space-dependant definition can be also presented. = 0 be a manifold lying locally in the vicinity Definition 9.2. Let ψ(X) = 0 for a flow of a manifold φ(X) = 0. A local invariant manifold φ(X) n Φt : D → R associated with the dynamical system (8.1) is such that: = k(X)φ( LV φ(X) X) MF 22

∈ φ and ∀X

∈ψ X

(9.3)

Example 9.2. Local invariance Let’s consider the following dynamical system: dX dt

dx

dt dy dt

=

f (x, y) g (x, y)

1 =

ε (y

− x2 )

−x

The flow curvature manifold associated with this system is: = φ(x, y) = φ(X) =− Its Lie derivative reads: LV φ(X) = T r(J) = − By posing k(X)

x4 − y 2 − x2 ε ε2

2x x4 − y 2 − x2 ε 2x(x2 − y)2 − ε ε2 ε3

2 2 2x = 2x(x − y) we have: and ψ(X) ε ε3

= k(X)φ( + ψ(X) LV φ(X) X) Thus, as soon as φ lies in the vicinity of ψ the flow curvature manifold ψ(X) and L φ(X) is locally invariant. In Fig. 9.1 where manifolds φ(X), V have been plotted in blue, green and magenta respectively it can be noticed that both manifolds φ and LV φ are merged as soon as they are lying in the vicinity of ψ.

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Y 1.5

1.0

0.5

0.0

X

0.5

1.0

1.5 1.0

0.5

Fig. 9.1

9.2

0.0

0.5

1.0

Local invariance

Linear invariant manifolds

Proposition 9.1. Any linear invariant manifold is in factor of the flow curvature manifold. Proof.

= a0 + a1 x1 + a2 x2 + . . . + an xn = a0 + Let ϕ(X)

n $

ai xi be a

i=1

is invariant. n-dimensional hyperplane. Let’s suppose that ϕ(X) Thus, according to (9.1): = k1 (X)ϕ( = a1 x˙ 1 + a2 x˙ 2 + . . . + an x˙ n = LV ϕ(X) X)

n

ai x˙ i

(9.4)

i=1

˙ X) = k2 (X), time-derivative of (9.4) leads to: While setting: (k 2 + k)( ˙ X)ϕ( = (k 2 + k)( = k2 (X)ϕ( = LV (LV ϕ(X)) X) X)

n i=1

ai xï

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Then, applying the Leibniz formulae the nth time derivative reads:

= LnV ϕ(X)

n−1

Cnp

p=0

p dn−p−1 d ϕ(X) k( X) dtn−p−1 dtp

= = kn (X)ϕ( X)

n

(n)

ai xi

(9.5)

i=1

The proof is based on the well-known determinant properties: P1 : Linear combinations of rows or columns leave the determinant unchanged. P2 : Multiplying a row or column by m multiplies the determinant by m. This follows from the multiplicative property and the determinants of the elementary matrix transformation matrices. The flow curvature manifold (7.18) reads: 

x˙ 1

x (n)  ¨1 = det  . = det X, ˙ X, ¨ X, ..., X φ(X)  ..  (n) x1

 x˙ 2 · · · x˙ n x¨2 · · · x ¨n   .. ..  . ··· .   (n) (n) x2 · · · xn

(9.6)

While replacing the last column by a linear combination of all the others, and taking into account Eqs. (9.4), (9.5) and (P1 ) property we have: 

   x  ¨1 = det  φ(X)  .  ..   (n) x1

n $



  x˙ 1    x x ¨2 · · · ai xï    ¨1  = det  i=1   ..  . .. ..   . ··· .  (n) n  $ (n) (n) x1 x2 · · · ai x

x˙ 1 x˙ 2 · · ·

ai x˙ i

i=1 n $

i=1

i

x˙ 2 · · · k1 (X)ϕ( X)



  x ¨2 · · · k2 (X)ϕ( X)    .. ..  . ··· .  (n) x2 · · · kn (X)ϕ(X)

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(P2 ) property leads to: 

x˙ 1 x˙ 2 · · · k1 (X)

 x  ¨1  = ϕ(X)det φ(X)  .. .  (n) x1



  x ¨2 · · · k2 (X)    .. ..  . ··· .  (n) x2 · · · kn (X)

is in factor of the flow curvature So, the linear invariant manifold ϕ(X) manifold. Remark. Such result may be extended to n linear invariant manifolds in factor of the flow curvature manifold. Nevertheless, any manifold in factor of the flow curvature manifold is not necessarily invariant. In a previous publication Jaume Llibre (Llibre and Zhang, 2004) has established a similar result with a method based on the use of extatic algebraic manifolds discovered by Mikhail Nikolaevich Lagutinskii in the beginning of the XXth century (Dobrovol’skii et al., 1998) and re-discovered by Arnold (1984), Jaume Llibre, Colin Christopher et Jorge Vitorio Pereira (Christopher et al., 2007). Thus, it is possible to show that linear invariant manifolds are extatic algebraic manifolds (Cf. supra). Proposition 9.2. The product of invariant manifolds is also an invariant manifold. and ϕ2 (X) and their Proof. Let’s consider two invariant manifold ϕ1 (X) product φ(X) = ϕ1 (X)ϕ2 (X). Its Lie derivative reads: = L ϕ1 (X) ϕ2 (X) + ϕ1 (X) L ϕ2 (X) LV φ(X) V V with i = 1, 2 are invariant manifolds we have according to Since ϕi (X) Def. 9.1: = ki (X)ϕ i (X) with i = 1, 2 LV ϕi (X)

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So, the Lie derivative of φ may be written: = k1 (X)ϕ 1 (X)ϕ 2 (X) + k2 (X)ϕ 1 (X)ϕ 2 (X) LV φ(X) + k2 (X))ϕ = (k1 (X) 1 (X)ϕ2 (X) + k2 (X))φ( X) = (k1 (X) by posing This proof may be extended by recurrence to n factors ϕi (X) n , and according to the Lie derivative properties. = ϕi (X) φ(X) i=1

MF 23

Example 9.3. Volterra-Lotka predator-prey model

V

x˙ f (x, y) x (3 − x − 5y) = = y˙ g (x, y) y (−1 + x + y)

The flow curvature manifold associated with this model reads: = xy (−3 + x + 3y) Q(X) =0 φ(X) is a polynomial of degree two. Thus, according to Prop. 9.1 where Q(X) factors x, y and (−3 + x + 3y) are linear invariant manifolds for this model. = xy(−3 + x + 3y) the Lie derivative of these factors reads: By posing ϕ(X) = xy (−3 + x + 3y) (2 − x − 3y) = k(X)ϕ( X) LV ϕ(X) = T r (J) = (2 − x − 3y). Thus, this Volterra-Lotka predatorwith k(X) prey model has three linear invariant manifolds. Moreover, second degree algebraic invariant manifolds may also be in factor of the flow curvature manifold but are nothing but decomposable conics (resp. quadrics) as highlighted in the following Ex. 9.4.

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Example 9.4. Pendulum

V

x˙ y f (x, y) = = y˙ g (x, y) −x

The flow curvature manifold associated with this system reads: = x2 + y 2 = 0 φ(X) = (x + iy) (x − iy) = 0. Thus, according and may also be written: φ(X) to Prop. 9.1 manifolds: ϕ (x, y) = x + iy and ϕ¯ (x, y) = x − iy are linear manifolds invariant with respect to the flow of this system. Then, let’s compute de Lie derivative of each of these manifolds: = 1x˙ + iy˙ = y − ix = −i (x + iy) = −iϕ(X) with k(X) = −i LV ϕ(X)

¯ X) = 1x˙ − iy˙ = y + ix = i (x − iy) = iϕ( with k( =i LV ϕ( ¯ X) ¯ X) Thus, the pendulum system has two linear invariant manifolds. Then, the converse to Prop. 9.1 may also be established by using the following theorems inherent to Differential Geometry: Theorem 1 : (t) is a straight A necessary and sufficient condition that the curve X ∧ γ = 0. line is that V Theorem 2 : A necessary and sufficient condition that the curvature is identically zero is that the curve be a straight line. Theorem 3 : A necessary and sufficient condition that the torsion is identically zero is that the curve be a plane curve. Proof.

Cf. Graustein (1935, p. 27, 32, 36 and next).

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Converse proposition 9.1 Any n-dimensional dynamical system for which the curvature of the trajectory curve is identically zero admits at least one linear invariant manifold. Proof. Two-dimensional dynamical systems If the curvature manifold of a two-dimensional dynamical system is identically zero, then, according to Serret-Frénet formulae (7.14): V dτ = ν = 0 dt a , where a is a a non null constant vector. Since, according to Eq. (7.11) the unit tangent

vector is defined as: τ = VV while supposing that V has a unit speed, i.e. + + + + +V + = 1 we have: Thus, by integrating this equation we have: τ =

dX a = dt a Integrating again leads to: X = at + b, with b is a constant vector, that is a straight line. τ = V =

Example 9.5. Emphasizing the proof According to Th. 7.3 and Def. 7.2, the 1st flow curvature manifold of a two-dimensional dynamical system reads: x˙ y˙ = x¨ κ1 = 0 ⇔ ⇔ φ(X) = ˙ y − y˙ x¨ = 0 x ¨ y¨ x ¨ y¨ = . By integrating we This last expression may be written as: y˙ x˙ obtain: Ln (y) ˙ = Ln (x) ˙ + C where C is a constant. It leads to y˙ = ax˙ where a = eC is a constant. Integrating again provides: y = ax + b where b is a constant, i.e. the equation of a straight line. + + =+ + φ(X) +γ ∧ V +=0

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Three-dimensional dynamical systems If the flow curvature manifold of a three-dimensional dynamical system is identically zero, then, according to Serret-Frénet formulae (7.14): V dβ = ν = 0 dt

d 0 = dX · β 0 = τ · β0 . ·β Hence, β = β0 is a constant vector. So, dt X dt d 0 = 0. Integrating again, ·β X Since, τ and β0 are orthogonal, then dt 0 = K, ·β with K is a constant vector, that is X is a plane. we have X MF 24

Example 9.6. Volterra-Lotka predator-prey model Let’s consider the polynomial dynamical system of Ex. 6.12:       x˙ f (x, y, z) x(z − y)  g (x, y, z)  =  y(x + z)   y˙  = V z˙ h (x, y, z) z(x + y) The flow curvature manifold associated with this system is identically zero. Thus, according to converse proposition 9.1 this dynamical system admits at least a linear invariant manifold . It may be checked that, in addition to x, y and z, x + y and y − z are linear invariant manifolds for this system2 . Remark. The proof for a two-dimensional dynamical system is obvious according to Th. 1 since when the flow curvature manifold of a twodimensional dynamical system is identically zero, then the instantaneous velocity and acceleration vectors are collinear. This is a necessary and sufficient condition that the trajectory curve is a straight line. Moreover, it will be established in the following that indeed any dynamical system of dimension two (resp. three) for which the curvature of the trajectory curve is identically zero admits a linear invariant manifold as first integral (Cf. Ch. 12). It is also important to note that Prop. 9.1 is only a sufficient 2 It

will be established in Ch. 12 that x+y −z is first integral for this dynamical system.

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and not a necessary and sufficient condition for a dynamical system to have linear algebraic invariant manifolds, although it has been possible to establish the converse to Prop. 9.1. 9.3

Nonlinear invariant manifolds

Analytical computation of invariant manifold of degree n, implies the use of the notion of extatic algebraic manifolds discovered by Mikhail Nikolaevich Lagutinskii at the beginning of the twentieth century Dobrovol’skii et al. (1998) and re-discovered by Arnold (1996), Jaume Llibre, Colin Christopher and Jorge Vitorio Pereira (Christopher et al., 2007). In this section it will be established on the one hand that extatic algebraic manifolds enables to compute higher degree invariant manifold analytical equation and, on the other hand, that flow curvature manifolds are extatic algebraic manifolds. Proposition 9.3. Any algebraic manifold of degree3 d (d 1) invariant with respect to the flow of a dynamical system, is in factor of the dth extatic algebraic manifold. Proof. First, let’s define, according to Jaume Llibre and Joao Medrado given (LLibre and Medrado, 2007), the dth extatic algebraic manifold εd (X) by the determinant of a matrix such that the first row is formed by a basis B of the monomials in x, y and z of degree d (d 1), and the ith row is the th Lie derivative LV applied in the (i − 1) row and where B = {v1 , v2 , ..., vl } with n = dim (B).  v2 ··· vn   L  V (v1 ) LV (v2 ) · · · LV (vn )    εd (X) =  .. .. ..  . . ··· .   n−1 n−1 n−1 LV (v1 ) LV (v2 ) · · · LV (vn ) 

v1

(9.7)

Let ϕ(X)be an invariant algebraic manifold of degree d. Since the choice of the basis plays no role in the definition of extatic algebraic manifold, we appears. Since can choose a basis where ϕ(X) = k1 (X)ϕ( X) LV ϕ(X) 3 The

case d = 1 has been considered in the previous section.

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˙ X)ϕ( = (k 2 + k)( = k2 (X)ϕ( L2V ϕ(X) X) X) While using the so-called Leibniz formulae it can be established that: = LnV ϕ(X)

n−1 p=0

Cnp

p dn−p−1 d ϕ(X) = kn (X)ϕ( k( X) X) dtn−p−1 dtp

is a polynomial. It is obvious that ϕ(X) is a factor of εd (X). where k(X) This proof is quite similar to those due to J. V. Pereira (Pereira, 2001) and provided in the previous sections. MF 25

Example 9.7. Non decomposable quadrics Let’s consider the dynamical system (Pikovskii et al., 1978, p. 716):       x˙ f (x, y, z) hy − ν1 x − yz  g (x, y, z)  = hx − ν2 y + xz  y˙  = V xy − ν3 z z˙ h (x, y, z) If ν1 = ν2 = ν3 = ν it has been established (Bountis et al., 1984; Giacomini et al., 1991) that this system admits the non-decomposable quadrics x2 − y 2 + 2z 2 as 2nd degree algebraic invariant manifold. According to Prop. 9.3 the 2nd extatic algebraic manifold reads: 

 x2 y2 z2   = LV x2 LV y 2 LV z 2  = 4νx2 y 2 (xy − νz) x2 − y 2 + 2z 2 εd (X)   L2V x2 L2V y 2 L2V z 2 = x2 − y 2 + 2z 2 its Lie derivative may be written: By posing ϕ(X) = ν(x2 − y 2 + 2z 2 ) = k1 (X)ϕ( LV ϕ(X) X) = ν. So, x2 − y 2 + 2z 2 is a 2nd degree algebraic invariant where k1 (X) manifold for this dynamical system. Moreover, let’s note that neither x2 y 2 nor (xy − νz) are invariant algebraic manifolds as it may be checked by computing their Lie derivative. Thus, Prop. 9.3 is also only sufficient and not a necessary and sufficient condition for a dynamical system to have dth degree algebraic invariant manifolds.

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Now, let’s prove the following proposition: Proposition 9.4. Flow curvature manifolds are extatic algebraic manifolds. given by Proof. Let’s consider the dth extatic algebraic manifold εd (X) the determinant of a matrix such that the first row is formed by a basis B where B = {x˙ 1 , x˙ 2 , ..., x˙ n } with n = dim (B). 

 x˙ 2 ··· x˙ n  LV (x˙ 2 ) · · · LV (x˙ n )      εd (X) =  .. ..  . ··· .   n−1 n−1 n−1 LV (x˙ 1 ) LV (x˙ 2 ) · · · LV (x˙ n ) x˙ 1 LV (x˙ 1 ) .. .

(9.8)

(n)

Since LV (x˙ 1 ) = x ¨1 , LV (x˙ 1 ) = x ¨1 and Ln−1 (x˙ i ) = xi determinant (9.8)

V reads: 

x˙ 1 x  ¨1 = . εd (X)  ..  (n) x1

 x˙ n x¨n   ..  .   (n) (n) x2 · · · xn x˙ 2 · · · x ¨2 · · · .. . ···

(9.9)

A simple comparison between this expression (9.9) and those for the flow curvature manifold (9.6) directly shows the identity. Remark. If both methods lead to the same results, the approach used is quite different. Flow curvature method involves a deductive reasoning since the existence of linear invariant manifold is deduced from the fact that it is a factor of the flow curvature manifold. Lagutinskii’s method makes use of inductive reasoning, because the basis is formed in function of the manifold searched. Nevertheless, except the case of decomposable quadrics, flow curvature manifolds can only provide linear invariant manifolds. But, it will be established in Ch. 12 (Cf. supra) that the existence of linear invariant manifolds in polynomial dynamical systems enables to study integrability of such systems while using Darboux theory of integrability.

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Chapter 10

Local Bifurcations

“If that enabled us to predict the succeeding situation with the same approximation, that is all we require. . . ” — H. Poincaré —

In the first part of this work (Ch. 4) it has been recalled that the center manifold theorem enables to study local bifurcations of codimension 1 such as saddle-node, transcritical, pitchfork or Hopf bifurcations. In this chapter it will be established that both center manifold and normal forms of any n-dimensional dynamical system may be directly deduced from the flow curvature manifold. 10.1 10.1.1

Center Manifold Center manifold approximation

Let’s consider a system of differential equations defined in a compact E = [x1 , x2 , ..., xn ]t ∈ E ⊂ Rn : included in R with X dX X) = ( dt

(10.1)

t X) = f1 (X), f2 (X), ..., fn (X) where ( ∈ E ⊂ Rn defines in E a velocity vector field whose components fi , supposed to be C ∞ continuous functions in E with values in R, are checking the assumptions of the Cauchy-Lipshitz theorem.

159

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Let’s rewrite1 dynamical system (10.1) as:

x˙ = Ax + f (x, y ) y˙ = By + g (x, y)

(10.2)

where x ∈ Rn , y ∈ Rm and A is a n × n constant matrix such that all its eigenvalues have zero real parts while B is a m × m constant matrix such that all its eigenvalues have negative real parts. Functions f and g are C 2 with f(0, 0) = 0, DX f(0, 0) = 0, g(0, 0) = 0, DX g(0, 0) = 0 and x, y ) so that D represents the functional jacobian matrix. When the X( X functional Jacobian matrix is diagonalizable dynamical system (10.1) may be transformed into (10.2) by using the eigenbasis. With (x, y ) ∈ Rn × Rm , the n-dimensional center manifold reads: y = h (x). Then, according to Def. 7.2 the flow curvature manifold associated with system (10.2) reads: ˙ · =X φ(X)

(n) ... ¨ ∧ X ∧ ...∧ X X

(n) ... ¨ X, ..., X ˙ X, = det X,

=0

(10.3)

(n)

represents the time derivatives of X = [x1 , x2 , ..., xn ]t . where X Proposition 10.1. The center manifold associated with any n-dimensional dynamical system is a polynomial whose coefficients may be deduced from the flow curvature manifold. Proof. The proof is based on the concept of invariant manifolds (Cf. Ch. 9). Two-dimensional dynamical systems According to Th. 7.3 and Def. 7.2 the 1st flow curvature manifold of a two-dimensional dynamical system reads: + + + ˙ ¨ + + +X ∧ X κ1 = + +3 + ˙ + +X + 1 For

⇔

+ + ¨ + ˙ ∧ X =+ φ(X) +=0 +X

sake of simplicity arrows may be forgotten.

⇔

φ (x, y) = 0

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Writing the total differential of the 1st flow curvature manifold leads to: ∂φ ∂φ dx + dy = 0 ∂x ∂y

dφ (x, y) =

(10.4)

But, according to the Center Manifold Theorem (Ch. 4, Th. 4.1) the center manifold y = h (x) solution of the partial equation (4.4) can be approximated arbitrarily closely by a Taylor series at x = 0:

y = h (x) =

n

ap0 xp = a20 x2 + a30 x3 + O x4

(10.5)

p=0

∂h dx By replacing in Eq. (10.4) dy by its total differential, i.e. by dy = ∂x yields dφ(x, y) =

∂φ ∂φ ∂h + ∂x ∂y ∂x

dx

(10.6)

So, according to Def. 9.1 φ(x, y) is globally invariant iff dφ(x, y) = 0, i.e. if ∂φ ∂φ ∂h + =0 ∂x ∂y ∂x

⇔

∂φ ∂h ∂x = − ∂φ ∂x ∂y

(10.7)

Replacing y = h (x) with expression (10.5) in both parts of Eq. (10.7) leads to: ∂h = h (x) = 2a20 x + 3a30 x2 + O x3 = − ∂x

By posing: a10 = −

∂φ ∂x ∂φ ∂y

∂φ ∂x ∂φ ∂y y=h(x)

(10.8)

derivative of Eq. (10.8) with respect to

y=h(x)

x reads:  d y ∂  = h (x) = 2a20 + 6a30 x + O x2 = − dx2 ∂x 2



∂φ ∂x  ∂φ ∂y y=h(x)

=

∂a10 ∂x

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By using a recurrence reasoning it may be stated that the unknown coefficients of the center manifold associated with a two-dimensional dynamical system: y = h (x) =

n

ap0 xp = a20 x2 + a30 x3 + O x4

p=0

are given by the following expressions: a20 = lim

x→0

a30 = lim

x→0

1 ∂a10 2! ∂x

1 ∂ 2 a10 3! ∂x2

··· an+1,0 = lim

x→0

1 ∂ n a10 (n + 1)! ∂xn

with n ≥ 1

(10.9)

MF 26

Example 10.1. Center manifold 2D Let’s consider the dynamical system of Ex. 4.2 (Guckenheimer and Holmes, 1983, p. 133): V

x˙ xy f (x, y) = = y˙ g (x, y) −y + αx2

The center manifold solution of Eq. (4.4) may be approximated arbitrarily closely as a Taylor series at x = 0: y = h (x) = a20 x2 + a30 x3 + a40 x4 + a50 x5 + O x6 where the coefficients are: a20 = α, a30 = a50 = 0, a40 = −2α2 . The 1st flow curvature manifold associated with this system reads: φ (x, y) = x y 3 + αx2 y + αx2 y 2 − α2 x4

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According to Eq. (10.9):

1 ∂a10 5α − 3a20 leads to a20 = α = 2! ∂x 2

1 ∂ 2 a10 = −6a30 leads to a30 = 0 3! ∂x2

1 ∂ 3 a10 = −84α2 −18a40 leads to a40 = −2α2 4! ∂x3

1 ∂ 4 a10 = −72a50 leads to a50 = 0 5! ∂x4

a20 = lim

x→0

a30 = lim

x→0

a40 = lim

x→0

a50 = lim

x→0

An alternative method may consist in plugging into the flow curvature manifold the expression of the center manifold and by identifying each power in order to find unknown coefficients. Thus, solving the following system order by order:

φ(x, y) = x y 3 + αx2 y + αx2 y 2 − α2 x4

y = h (x) = a20 x2 + a30 x3 + a40 x4 + a50 x5 + O x6

also yields to the unknown coefficients: a20 = α, a30 = a50 = 0, a40 = −2α2 So, the center manifold equation plotted on Fig.10.1 reads: y = h (x) = αx2 − 2α2 x4 + O x6 Thus, the reduced system is x˙ = αx3 − 2α2 x5 + O x7

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Y 4

2

0

X

2

4 4

2

Fig. 10.1

0

2

4

Center manifold in green for α = −0.25.

Three-dimensional dynamical systems According to Th. 7.4 and Def. 7.2 the 2nd flow curvature manifold of a three-dimensional dynamical system reads: ... ¨ ∧ X ˙ · X X κ2 = + +2 + ˙ ¨ + + +X ∧ X

⇔

... ˙ ¨ φ(X) = X· X ∧ X = 0

⇔

φ (x, y, z) = 0

Writing the total differential of the 2nd flow curvature manifold leads to: dφ (x, y, z) =

∂φ ∂φ ∂φ dx + dy + dz = 0 ∂x ∂y ∂z

(10.10)

According to the Center Manifold Theorem (Ch. 4, Th. 4.1) the center manifold z = h (x, y) solution to the partial equation (4.4) can be approximated arbitrarily closely as a Taylor series at (x, y) = (0, 0): z = h (x, y) =

n $ p=0

a2−p,p x2−p y p +

n $ p=0

a3−p,p x3−p y p + O x4 , y 4

= a20 x2 + a11 xy + a02 y 2 + a30 x3 + a21 x2 y + a12 xy 2 + a03 y 3

(10.11)

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By replacing in Eq. (10.10) dz by its total differential dz =

∂h ∂h dx+ dy ∂x ∂y

yields dφ(x, y, z) =

∂φ ∂φ ∂h + ∂x ∂z ∂x

dx +

∂φ ∂φ ∂h + ∂y ∂z ∂y

dx

(10.12)

According to Def. 9.1 φ(x, y, z) is globally invariant iff dφ(x, y, z) = 0, i.e. if  ∂φ ∂φ ∂h    + =0   ∂x ∂z ∂x   ∂φ ∂φ ∂h   + =0  ∂y ∂z ∂y

∂φ

⇔

∂h ∂x = − ∂φ ∂x ∂z

⇔

∂h ∂y = − ∂φ ∂y ∂z

∂φ

(10.13)

By replacing in both parts of Eq. (10.13) z = h (x, y) by its expression (10.11) and by setting:

a10 = −

∂φ ∂x ∂φ ∂z z=h(x,y)

and

a01 = −

∂φ ∂y ∂φ ∂z z=h(x,y)

leads to: = a10 z=h(x,y) ∂φ ∂y = a11 x + 2a02 y + O x2 , y 2 = − ∂φ = a01 ∂z

 2 2 ∂h(x,y)   ∂x = 2a20 x + a11 y + O x , y = −   ∂h(x,y)   ∂y

∂φ ∂x ∂φ ∂z

(10.14)

z=h(x,y)

By using a a recurrence reasoning it may be stated that the unknown coefficients of the center manifold associated with a three-dimensional dynamical system: z = h (x, y) = a20 x2 +a11 xy+a02 y 2 +a30 x3 +a21 x2 y+a12 xy 2 +a03 y 3 +· · · are given by the following expressions: an+1,0 =

lim

(x,y)→(0,0)

a0,n+1 =

lim

(x,y)→(0,0)

1 ∂ n a10 (n + 1)! ∂xn n

1 ∂ a01 (n + 1)! ∂y n

with n ≥ 1 (10.15)

with n ≥ 1

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The coefficient of xy reads a11 =

lim

(x,y)→(0,0)

∂a10 ∂a01 = lim ∂y ∂x (x,y)→(0,0)

MF 27

Example 10.2. Center manifold 3D Consider the dynamical system of Ex. 4.3 (Dang-Vu, 2000, p. 40):       x˙ f (x, y, z) −y + xz  g (x, y, z)  =   V  y˙  = x2+ yz2 2 z˙ h (x, y, z) −z − x + y + z The center manifold z = h (x, y) solution to the partial equation (4.4) may be approximated arbitrarily closely as a Taylor series at (x, y) = (0, 0): z = h (x, y) = a20 x2 +a11 xy+a02 y 2 +a30 x3 +a21 x2 y+a12 xy 2 +a03 y 3 +· · · where coefficients are: a20 = a02 = −1, a11 = a30 = a03 = a21 = a12 = 0 The 2nd flow curvature manifold associated with this system reads: φ (x, y, z) = 0 (Cf. Mathematica File). According to Eqs. (10.15):

1 ∂a10 = − (2 + a20 ) leads to a20 = −1 2! ∂x (x,y)→(0,0) 1 ∂a01 = lim = − (2 + a02 ) leads to a02 = −1 2! ∂y (x,y)→(0,0) ∂a10 ∂a01 = lim = lim = a11 leads to a11 = 0 (x,y)→(0,0) ∂y (x,y)→(0,0) ∂x

a20 = a02 a11

lim

It may also be stated that: a30 = a03 = a21 = a12 = 0 So, the center manifold equation reads: z = h (x, y) = −x2 − y 2 + O x4 , y 4

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Remark. These results may be extended to n-dimensional dynamical systems. Moreover, an alternative proof may be stated by plugging into the flow curvature manifold the expression of the center manifold and by identifying each power in order to find the unknown coefficients. 10.1.2

Center manifold depending upon a parameter

Let’s consider a differential equations system defined in a compact E in = [x1 , x2 , ..., xn ]t ∈ E ⊂ cluded in R depending on a parameter µ with X Rn , µ ∈ R: dX X, µ) = ( dt

(10.16)

X) = [f1 (X), f2 (X), ..., fn (X)] t ∈ E ⊂ Rn defines in E a velocity where ( vector field whose components fi , supposed to be C ∞ continuous functions in E with values in R, are checking the assumptions of the Cauchy-Lipshitz theorem. Dynamical system (10.16) depending upon a vector parameter µ may be rewritten as:    x˙ = Ax + f (x, y, µ) y˙ = By + g (x, y , µ)   µ˙ = 0

(10.17)

with (x, y, µ) ∈ Rn × Rm × Rp . The (n + p)-dimensional center manifold reads y = h (x, µ ). Then, according to Def. 7.2 the flow curvature manifold associated with system (10.17) reads: µ) = X ˙ · φ(X,

(n) ... ∧ ...∧ X ¨ ∧ X X

(n) ... ..., X ˙ X, ¨ X, = det X,

= 0 (10.18)

(n)

represents the time derivatives of X = [x1 , x2 , ..., xn ]t . where X Proposition 10.2. The center manifold associated with any n-dimensional dynamical system is a polynomial whose coefficients may be deduced from the flow curvature.

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Proof. The proof is still based on the concept of invariant manifolds (Cf. Ch. 9). Two-dimensional dynamical systems According to Th. 7.3 and Def. 7.2 the flow curvature manifold of a two-dimensional dynamical system reads: + + + ˙ ¨ + + +X ∧ X κ1 = + +3 + ˙ + +X +

⇔

+ + ˙ ∧ X ¨ + µ) = + φ(X, +=0 +X

⇔

φ (x, y, µ) = 0

Writing the total differential of the 1st flow curvature manifold leads to: dφ (x, y, µ) =

∂φ ∂φ ∂φ dx + dy + dµ = 0 ∂x ∂y ∂µ

(10.19)

According to the Center Manifold Theorem (Ch. 4, Th. 4.1) the center manifold y = h (x, µ) solution of the partial equation (4.4) can be approximated arbitrarily closely as a Taylor series at (x, µ) = (0, 0): y = h (x, µ) =

n $ p=0

a2−p,p x2−p µp +

n $ p=0

a3−p,p x3−p µp + O x4 , µ4

(10.20)

= a20 x2 + a11 xµ + a02 µ2 + a30 x3 + a21 x2 µ + a12 xµ2 + a03 µ3 Replacing in Eq. (10.19) dy by its total differential dy =

∂h ∂h dx + dµ ∂x ∂µ

yields dφ(x, y, µ) =

∂φ ∂φ ∂h + ∂x ∂y ∂x

dx +

∂φ ∂φ ∂h + ∂µ ∂y ∂µ

dµ

(10.21)

According to Def. 9.1 φ(x, y, µ) is globally invariant iff dφ(x, y, µ) = 0, i.e. if    ∂φ + ∂φ ∂h = 0    ∂x ∂y ∂x   ∂φ ∂φ ∂h   + =0  ∂µ ∂y ∂µ

∂φ

⇔

∂h ∂x = − ∂φ ∂x ∂y

⇔

∂h ∂µ = − ∂φ ∂µ ∂y

∂φ

(10.22)

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By replacing y = h (x, µ) by its expression (10.20) in both parts of Eq. (10.22) and by setting

a10 = −

∂φ ∂x ∂φ ∂y y=h(x,µ)

and

a01 = −

∂φ ∂µ ∂φ ∂y y=h(x,µ)

leads to: = a10 y=h(x,µ) ∂φ = a11 x + 2a02 µ + O x2 , µ2 = − ∂µ = a01 ∂φ ∂y

 2 2 ∂h(x,µ)    ∂x = 2a20 x + a11 µ + O x , µ = −  ∂h(x,µ)   ∂µ

∂φ ∂x ∂φ ∂y

(10.23)

y=h(x,µ)

By using a recurrence reasoning it may be stated that the unknown coefficients of the center manifold associated with a two-dimensional dynamical system: y = h (x, µ) = a20 x2 + a11 xµ+ a02 µ2 + a30 x3 + a21 x2 µ+ a12 xµ2 + a03 µ3 + · · · are given by the following expressions: an+1,0 =

lim

(x,µ)→(0,0)

a0,n+1 =

lim

(x,µ)→(0,0)

1 ∂ n a10 (n + 1)! ∂xn n

1 ∂ a01 (n + 1)! ∂µn

The coefficient of xµ reads a11 =

lim

(x,µ)→(0,0)

with n ≥ 1 (10.24)

with n ≥ 1

∂a10 ∂a01 = lim ∂µ (x,µ)→(0,0) ∂x

MF 28

Example 10.3. Center manifold 2D depending upon a parameter. Let’s consider the Quadratic Duffing dynamical system: V

x˙ y f (x, y) = = y˙ g (x, y) µx − x2 − δy

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170

In its eigenbasis this system reads:

2 µ 1 x ˙ f (x, y) (x + y) − ( x + y) δ δ V = = 2 y˙ g (x, y) −δy − µδ (x + y) + 1δ ( x + y) The center manifold solution of Eq. (4.4) may be approximated arbitrarily closely as a Taylor series at (x, y) = (0, 0) and at µ = 0: y = h (x, µ) = a20 x2 + a11 xµ + a02 µ2 + O x3 , µ3 where the coefficients are: a20 =

1 1 , a11 = − 2 and a02 = 0. δ2 δ

The 1st flow curvature manifold associated with this system reads:

φ (x, y, µ) =

(x + y)2 (x + y − µ)2 + y y 2 + x (−x + µ) δ δ

According to Eqs. (10.24):

a20

1 ∂a10 1 2 − a20 δ 2 = lim provides a20 = 2 = x→0 2! ∂x δ2 δ

a02

1 ∂a10 1 a02 provides a02 = 0 or a02 = − 2 = lim =− 2 x→0 2! ∂µ (2 + 3a02 δ ) δ

a11 =

lim

(x,µ)→(0,0)

provides a11 = −

∂a10 ∂a01 2 + (a20 + 2a11 ) δ 2 = lim =− ∂µ ∂x δ2 (x,µ)→(0,0)

1 δ2

So, the center manifold equation reads: y = h (x, µ) =

1 2 x − µx + O x3 , µ3 2 δ

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171

Plugging it into Duffing system in its diagonal form gives the vector field reduced to the center manifold: x˙ F (x, µ) = µ˙ 0

with

F (x, µ) =

1 µ µ µ 1− 2 x− 1 − 2 x2 δ δ δ δ

Thus, it may be stated that conditions (4.28) are checked, i.e. F (0, 0) = Fx (0, 0) = 0 ; Fµ (0, 0) = 0 ; Fxx (0, 0) = −

2 = 0 δ

So, a transcritical bifurcation occurs for µ → 0. Three-dimensional dynamical systems According to Th. 7.4 and Def. 7.2 the 2nd flow curvature manifold of a three-dimensional dynamical system reads: ... ¨ ∧ X ˙ · X X ... ¨ ˙ κ2 = + ⇔ φ(X, µ) = X · X ∧ X = 0 ⇔ φ (x, y, z, µ) = 0 +2 + ˙ ¨ + + +X ∧ X Writing the total differential of the 2nd flow curvature manifold leads to:

dφ (x, y, z, µ) =

∂φ ∂φ ∂φ ∂φ dx + dy + dz + dµ = 0 ∂x ∂y ∂z ∂µ

(10.25)

According to the Center Manifold Theorem (Ch. 4, Th. 4.1) the center manifold solution of the partial equation (4.4) can be approximated arbitrarily closely as a Taylor series at (x, y, z, ε) = (0, 0, 0, 0): y = h1 (x, µ) = a20 x2 + a11 xµ + a02 µ2 + O x3 , µ3 z = h2 (x, µ) = b20 x2 + b11 xµ + b02 µ2 + O x3 , µ3

(10.26)

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Replacing in Eq. (10.25) dy by its total differential dy =

∂h1 ∂h1 dx+ dµ ∂x ∂µ

yields dφ(x, y, z, µ) =

∂φ ∂φ ∂h1 + ∂x ∂y ∂x

dx+

∂φ ∂φ ∂h1 ∂φ dz + + dµ (10.27) ∂z ∂µ ∂y ∂µ

So, according to Def. 9.1 φ(x, y, z, µ) is globally invariant iff dφ(x, y, z, µ) = 0, i.e. if z = 0 and  ∂φ ∂φ ∂h1   + =0    ∂x ∂y ∂x   ∂φ ∂φ ∂h1   + =0  ∂µ ∂y ∂µ

∂φ

⇔

∂h1 ∂x = − ∂φ ∂x ∂y

⇔

∂h1 ∂µ = − ∂φ ∂µ ∂y

(10.28)

∂φ

By replacing y = h1 (x, µ) by its expression (10.26) in both parts of Eq. (10.28) and by setting:

a10 = −

∂φ ∂x ∂φ ∂y y=h1 (x,µ), z=0

and

a01 = −

∂φ ∂µ ∂φ ∂y y=h1 (x,µ), z=0

leads to:   ∂h1 (x,µ)   = 2a20 x + a11 µ + O x2 , µ2 = −  ∂x       ∂h1 (x,µ)  = a11 x + 2a02 µ + O x2 , µ2 = −  ∂µ  

∂φ ∂x ∂φ ∂y y=h1 (x,µ) z=0 ∂φ ∂µ ∂φ ∂y y=h1 (x,µ) z=0

= a10 (10.29) = a01

Replacing in Eq. (10.25) dz by its total differential, i.e. by dz = dφ(x, y, z, µ) =

∂φ ∂φ ∂h2 + ∂x ∂z ∂x

∂h2 ∂h2 dx + dµ ∂x ∂µ

∂φ ∂φ ∂h2 ∂φ dy + + dx+ dµ (10.30) ∂y ∂µ ∂z ∂µ

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So, according to Def. 9.1 φ(x, y, z, µ) is globally invariant iff dφ(x, y, z, µ) = 0, i.e. if y = 0 and  ∂φ ∂φ ∂h2    + =0   ∂x ∂z ∂x   ∂φ ∂φ ∂h2   + =0  ∂µ ∂z ∂µ

∂φ

⇔

∂h2 ∂x = − ∂φ ∂x ∂z

⇔

∂h2 ∂µ = − ∂φ ∂µ ∂z

(10.31)

∂φ

By replacing z = h2 (x, µ) by its expression (10.26) in both parts of Eq. (10.31) and by setting:

b10 = −

∂φ ∂x ∂φ ∂z y=0, z=h2 (x,µ)

and

b01 = −

∂φ ∂µ ∂φ ∂z y=0, z=h2 (x,µ)

leads to    ∂h2 (x,µ) = 2b20 x + b11 µ + O x2 , µ2 = −   ∂x      ∂h2 (x,µ)   = b11 x + 2b02 µ + O x2 , µ2 = −    ∂µ

∂φ ∂x ∂φ ∂z y=0 z=h2 (x,µ)

= b10

∂φ ∂µ ∂φ ∂z y=0 z=h2 (x,µ)

= b01

(10.32)

By using a a recurrence reasoning it may be stated that the unknown coefficients of the center manifold associated with a three-dimensional dynamical system: y = h1 (x, µ) = a20 x2 + a11 xµ + a02 µ2 + O x3 , µ3 z = h2 (x, µ) = b20 x2 + b11 xµ + b02 µ2 + O x3 , µ3 are given by the following expressions:

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an+1,0 a0,n+1

∂ n a10 1 = lim with n ≥ 1 (x,µ)→(0,0) (n + 1)! ∂xn 1 ∂ n a01 = lim with n ≥ 1 (x,µ)→(0,0) (n + 1)! ∂µn

bn+1,0 b0,n+1

1 ∂ n b10 = lim with n ≥ 1 (x,µ)→(0,0) (n + 1)! ∂xn 1 ∂ n b01 = lim with n ≥ 1 (x,µ)→(0,0) (n + 1)! ∂µn

(10.33)

The coefficient of xµ reads:

a11 b11

∂a10 ∂a01 = lim = lim (x,µ)→(0,0) ∂µ (x,µ)→(0,0) ∂x ∂b10 ∂b01 = lim = lim (x,µ)→(0,0) ∂µ (x,µ)→(0,0) ∂x

MF 29

Example 10.4. Center manifold 3D depending upon a parameter. Let’s consider the Chen’s system (Chang and Chen, 2006):       x˙ f (x, y, z) a (y − x)  g (x, y, z)  =  (c − a) x − xz + cy   y˙  = V z˙ h (x, y, z) xy − bz In the eigenbasis this system reads:       x˙ f (x, y, z) −2µx − 3µz − 2xy − 4yz  g (x, y, z)  =  −by + x2 + 3xz + 2z 3   y˙  = V −cz + µx + µz + xy + 2yz z˙ h (x, y, z) This system exhibits a pitchfork bifurcation in the vicinity of the origin when µ → 0. According to the Center Manifold Theorem (Ch. 4, Th. 4.1) the center manifold solution of the partial equation (4.4) can be approximated arbitrarily closely as a Taylor series at (x, y, z, µ) = (0, 0, 0, 0): y = h1 (x, µ) = a20 x2 + a11 xµ + a02 µ2 + O x3 , µ3 z = h2 (x, µ) = b20 x2 + b11 xµ + b02 µ2 + O x3 , µ3

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where the coefficients are: a02 = a11 = b20 = b02 = 0, a20 =

175

1 1 , b11 = . b c

The 2nd flow curvature manifold associated with this system reads: φ (x, y, z, µ) = 0 (Cf. Mathematica File). According to Eqs. (10.33): a02 = a11 = b20 = b02 = 0

1 ∂ n a10 1 −5 + 3ba20 provides a20 = =− 2b b (x,µ)→(0,0) (n + 1)! ∂xn 1 ∂b10 ∂b01 = lim = lim provides b11 = (x,µ)→(0,0) ∂µ (x,µ)→(0,0) ∂x c

a20 = b11

lim

Remark. An alternative proof may also be stated by using the Euler’s theorem for homogeneous polynomials. This approach more appropriated to high-dimensional dynamical systems will be emphasized in the Part III of this book (Cf. Ch. 16).

10.2

Normal Form Theorem

Let’s suppose that the center manifold approximation has been applied to dynamical system (10.16) which has been transformed into (10.17). Omitting explicit reference to the parameter µ and to the center manifold y = h (x, µ ) such system may be written as: x˙ = Ax + f (x)

(10.34)

t t x = [x1 , x2 , . . . , xn ] ∈ Rn , f (x) = [f1 (x) , f2 (x) , ..., fn (x)] ∈ E ⊂ Rn and A is a n × n constant matrix such that all its eigenvalues have zero real parts. The Taylor series of f (x) at the origin leads to: k+1 x˙ = Ax + F (2) (x) + F (3) (x) + · · · + F (k) (x) + O(|x| ) (k)

(k)

(k)

(10.35)

where F (k) (x) = [F1 (x) , F2 (x) , . . . , Fn (x)]t are homogeneous polynomial of degree k in x.

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Then, according to Def. 7.2 the flow curvature manifold associated with system (10.35) reads: φ (x) = x˙ ·

(n) ... ¨ ∧ x ∧ . . . ∧ x x

(n) ... ¨, x , . . . , x = det x˙ , x

=0

(10.36)

(n)

where x represents the time derivatives of x = [x1 , x2 , . . . , xn ]t . Using the identity established in the appendix the flow curvature manifold may be written: k+1 φ (x) = (n) (x)+ (n+1) (x)+ (n+2) (x) · · ·+ (k) (x)+O |x|

(10.37)

The first term of the right hand side of Eq. (10.37) which reads (n) ... ¨ ∧ A2 x ∧ . . . ∧ An−1 x ) where |A| = det (A) and (n) (x) = |A| x˙ · (Ax (k) (x) is an homogeneous polynomial of degree k in x corresponds to the linear part of dynamical system (10.35). In this case normal forms may be computed by building a sequence of nonlinear transformations (variable changes) which successively remove the nonlinear terms starting from the terms of degree n + 1. Proposition 10.3. By a sequence of analytic coordinate changes of the form:

x = y + ℘ (y ) ,

℘ (y) ∈ Hr ,

r = 2, 3, . . . , k

the flow curvature manifold (10.37) is transformed into: k+1 φ (x) = (n) (x) + (n+1) (x) + (n+2) (x) · · · + (k) (x) + O |x| such that by a suitable choice of ℘ (y ) the terms of degree smaller than k are unchanged by this transformation while the terms of degree k can be eliminated.

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Proof. system:

177

For sake of simplicity let’s consider a two-dimensional dynamical

x˙ = Ax + F (x)

(10.38)

where F (2) (x) = F (x) is a quadratic nonlinearity made of homogeneous polynomials of degree two and defined by: F (x) =

f20 x21 + f11 x1 x2 + f02 x22 f20 x21 + f11 x1 x2 + f02 x22

(10.39)

Time derivative of instantaneous velocity vector (10.38) reads: ¨ = Ax˙ + F˙ (x) x ˙ Using the Frchet derivative F = D x F x˙ leads to: x¨ = Ax˙ + D x F x˙

(10.40)

By plugging Eq. (10.38) into Eq. (10.40) and keeping only terms of degree smaller than three provides: ¨ = A2 x˙ + AF (x) + D x F Ax + D x F F + O |x|4 x

(10.41)

The flow curvature manifold associated with dynamical system (10.38) may read2 : φ(x) = A2 x ∧ Ax + A2 x ∧ F + AF ∧ Ax + D x F Ax ∧ Ax + O |x|4 Using the second identity (A.15) established in the appendix and by posing: (3) (x) = (x) = A2 x ∧ F + AF ∧ Ax + D x F Ax ∧ Ax 2 For

sake of simplicity Euclidean norm has been dropped.

(10.42)

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Flow curvature manifold may finally be written: 4 φ(x) = |A| (Ax ∧ x) + (x) + O |x|

(10.43)

where |A| (Ax ∧ x) is a polynomial of degree two which corresponds to the linear part of dynamical system (10.38) and (x) is an homogeneous polynomial of degree three representing its nonlinear part. So, in order to eliminate this nonlinear part, let’s introduce a coordinate change of the form: x = y + ℘ (y )

(10.44)

where ℘ (y ) are homogeneous polynomial of degree two defined by: ℘ (y) =

f˜20 y12 + f˜11 y1 y2 + f˜02 y22 f˜20 y12 + f˜11 y1 y2 + f˜02 y22

(10.45)

and whose coefficients are to be determined. Then, plugging Eq. (10.44) into Eq. (10.43) and keeping only terms of degree smaller than three provides: φ(y ) = |A| (Ay ∧ y ) + |A| (Ay ∧ ℘) + |A| (A℘ ∧ y ) + (y ) + O |y |4

(10.46)

So, (y ) can be eliminated if ℘ (y ) can be chosen such that: |A| (A℘ ∧ y − ℘ ∧ Ay ) = − (y )

(10.47)

Equation (10.47) is the analog of Eq. (4.15) established in Ch. 4. By plugging Eq. (10.39) into Eq. (10.42) and replacing x by y gives: (y ) = A30 y13 + A21 y12 y2 + A12 y1 y22 + A03 y23 with A30 = −λ1 (λ1 + λ2 )b20 A21 = +λ2 [(3λ1 − λ2 )a20 − 2λ1 b11 ] A12 = +λ1 [(λ1 − 3λ2 )b02 + 2λ2 a11 ] A03 = +λ2 (λ1 + λ2 )a02

(10.48)

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179

Then, plugging Eq.(10.45) into the right hand side of Eq. (10.47) gives |A| (A℘ ∧ y − ℘ ∧ Ay ) = B30 y13 + B21 y12 y2 + B12 y1 y22 + B03 y23

(10.49)

with B30 = λ1 λ2 (λ1 − λ2 )g˜20 B21 = λ1 λ2 (λ1 − λ2 )(f˜20 + g˜11 ) B12 = λ1 λ2 (λ1 − λ2 )(f˜11 + g˜02 ) B03 = λ1 λ2 (λ1 − λ2 )f˜02 Thus, by replacing Eq. (10.48) and Eq. (10.49) in both sides of Eq. (10.47) and identifying order by order leads to: λ1 + λ2 a02 f˜02 = λ2 − λ1 λ1 3λ1 − λ2 a20 2b11 − f˜20 + g˜11 = λ2 − λ1 λ1 λ2 − λ1 − 3λ2 b02 2a λ 11 1 + f˜11 + g˜02 = λ2 − λ1 λ2 − λ1 λ2 λ1 + λ2 b20 g˜20 = − λ2 − λ1 λ2 which may be written:

f˜ij g˜ij

(m1 ± 1)λ1 + (m2 ∓ 1)λ2 = (λ2 − λ1 )λi

aij bij

(10.50)

with i = 1 (resp. 2) for f˜ij (resp. g˜ij ) and m1 + m2 = 2 as previously defined in Ch. 4. Remark. Although this transformation enables to find the “normal form” of the flow curvature manifold associated with dynamical system (10.38) it does not provide the normal form of such system which may be obtained by another transformation presented in Ch. 4. So, it is interesting to establish a link between both transformations.

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180

Thus, comparing Eq. (4.18) of 4.6 and Eq. (10.50) leads to:

fij gij

λi (λ2 − λ1 ) = (m1 λ1 + m2 λ2 − λi ) [(m1 ± 1)λ1 + (m2 ∓ 1)λ2 ]

f˜ij g˜ij

(10.51)

with i = 1 (resp. 2) for fij (resp. gij ) and m1 + m2 = 2 as previously defined in Ch. 4. Example 10.5. Linking both transformations Let’s consider the following dynamical system (4.5): x˙ =

x˙1 x˙2

= F

f1 (x1 , x2 ) f2 (x1 , x2 )

=

3x1 − x22 x2

It may be checked that flow curvature manifold associated with this system reads: = φ(x1 , x2 ) = 6x1 x2 − 4x3 φ(X) 2 According to 4.5 it has been stated that since λ1 = 3 and λ2 = 1: m1 = 0 and m2 = 2. Then, by application of Eq. (10.50) we have: 2 (0 + 1)3 + (2 − 1)1 f˜02 = (−1) = (1 − 3)3 3 Eq. (10.51) yields:

f02 =

3(1 − 3) 2 =1 (0.3 + 2.1 − 3) [(0 + 1)3 + (2 − 1)1] 3

Thus, parameter f02 leading to the normal form of this system may be found again starting from the flow curvature manifold.

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10.3

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181

Local bifurcations of codimension 1

It has been established in the previous section that the center manifold F (x, µ) may also be found again from the flow curvature manifold. Thus, according to conditions upon the partial derivatives of F (x, µ) a saddlenode, a transcritical or a pitchfork bifurcation will occur (Cf. Ch. 4).

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Chapter 11


“Discoveries, small or great are never born of spontaneous generation. They always presuppose a soil seeded with preliminary knowledge and well prepared by labor, both conscious and subconscious.” — H. Poincaré —

The concept of invariant manifolds plays a very important role in the stability and structure of dynamical systems and especially for slow-fast dynamical systems or singularly perturbed systems. Since the beginning of the twentieth century it has been subject to a wide range of seminal research. The classical geometric theory developed originally by Andronov et al. (1937), Tikhonov (1948, 1952) and Levinson (1950) stated that singularly perturbed systems possess invariant manifolds in which trajectories evolve slowly and toward which nearby orbits contract exponentially in time (either forward and backward) in the normal directions. These manifolds have been called asymptotically stable (or unstable) slow manifolds. Then, Fenichel (1971, 1979) theory1 for the persistence of normally hyperbolic invariant manifolds enabled to establish the local invariance of slow manifolds that possess both expanding and contracting directions and which were labeled slow invariant manifolds. Thus, various methods have been developed in order to determine the slow invariant manifold analytical equation associated with singularly perturbed systems. The fundamental works of Wasow (1965), Cole (1968), O’Malley (1974, 1991) and Fenichel (1971, 1979) to name but a few, gave rise to the so-called Geometric Singular Perturbation Theory and the problem for finding the slow invariant manifold analytical equation turned into a regular perturbation problem in which one generally expected, according to O’Malley (1974, p. 78) and O’Malley (1991, p. 21), asymptotic validity of such expansion to breakdown. 1 independently

developed in Hirsch et al. (1977) 183

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DGeometry


Another method called: Tangent Linear System Approximation, developed by Rossetto et al. (1998), consisted in using the presence of a “fast” eigenvalue in the functional jacobian matrix of low-dimensional (2 and 3) slow-fast dynamical systems. Within the framework of application of the theorem of Tikhonov (1948, 1952), this method used the fact that in the vicinity of the slow manifold the eigenmode associated with the “fast” eigenvalue was evanescent. Thus, the Tangent Linear System Approximation method provided the slow manifold analytical equation to lowdimensional slow-fast dynamical systems according to the “slow” eigenvectors of the tangent linear system, i.e. according to the “slow” eigenvalues. Nevertheless, occurrence of complex eigenvalues prevented from expressing this equation explicitly. Moreover, starting from dimension five Galois Theory precludes from analytically computing eigenvalues associated with the functional jacobian matrix of a five-dimensional dynamical system. Also to solve this problem it was necessary to make such equation independent on the “slow” eigenvalues. This could be carried out by multiplying it by “conjugated” equations leading to a slow manifold analytical equation independent on the “slow” eigenvalues of the tangent linear system. Thus, it was established in (Ginoux and Rossetto, 2006) while using the framework of Differential Geometry that the resulting equation was identically corresponding in dimension two to the curvature (first curvature) of the flow and in dimension three to the torsion (second curvature). Then, this approach has been generalized to high-dimensional dynamical systems. It will be stated in this chapter that the flow curvature manifold defined as the location of the points where the curvature of the flow, i.e. the curvature of trajectory curves of any n-dimensional slow-fast dynamical system vanishes directly provides its slow manifold analytical equation the invariance of which is established according to Darboux invariance theorem.

11.1

Slow manifold of n-dimensional slow-fast dynamical systems

Let’s consider a n-dimensional singularly perturbed system or slow-fast dynamical system, i.e. a system of differential equations defined in a compact = [x1 , x2 , ..., xn ]t ∈ E ⊂ Rn : E included in R with X dX X) = ( dt

(11.1)

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t X) = f1 (X), f2 (X), ..., fn (X) where ( ∈ E ⊂ Rn defines in E a velocity vector field whose components fi , supposed to be C ∞ continuous functions in E with values in R, are checking the assumptions of the Cauchy-Lipshitz theorem (Cf. Def. 5.1 & 5.2). Then, according to Def. 7.2 the flow curvature manifold associated with system (11.1) reads: =X ˙ · φ(X)

(n) ... ∧ ...∧ X ¨ ∧ X X

(n) ... ..., X ˙ X, ¨ X, = det X,

=0

(11.2)

(n)

represents the time derivatives of X = [x1 , x2 , ..., xn ]t . where X Proposition 11.1. The flow curvature manifold of any n-dimensional slow-fast dynamical system directly provides its slow invariant manifold analytical equation. Proof. According to the Tangent Linear System Approximation and its generalization2 it has been established that the slow manifold equation of any n-dimensional slow-fast dynamical system may be written: = V . Yλ2 ∧ . . . ∧ Yλn = 0 φ(X)

(11.3)

where Yλi are the eigenvectors associated with the functional jacobian matrix J of the tangent linear system. In the framework of the Generalized Tangent Linear System Approximation the functional jacobian matrix associated with the slow-fast dynamical system (11.1) is supposed to be locally stationary: dJ =0 dt As a consequence, time derivatives of acceleration vectors reads: (n)

γ = J (n+1) V = J (n)γ

2 See

Appendix for details.

(11.4)

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Then, mapping the flow of the tangent linear system, i.e. its functional jacobian operator, J to the velocity vector field spanned on the eigenbasis Eq. (11.3) leads to: = γ = JV

n

ai J Yλi = a2 J Yλ2 + . . . + an J Yλn

i=2

. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . . = J (n−2) V

(n−2)

γ

=

n

ai J (n−2) Yλi = a2 J (n−2) Yλ2 + . . . + an J (n−2) Yλn

i=2

Using the eigenequation J Yλk = λk Yλk , these equations may be written: = γ = JV

n

ai λi Yλi = a2 λ2 Yλ2 + . . . + an λn Yλn

i=2

. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . . = J (n−2) V

(n−2)

γ

=

n

Yλ2 + . . . + an λn−2 Yλn ai J (n−2) Yλi = a2 λn−2 n 2

i=2

Under the assumptions of the Generalized Tangent Linear System Ap(n−2)

proximation, it is obvious that vectors V ,γ , . . . , γ spanned on the same eigenbasis Yλ2 , Yλ3 , . . . , Yλn are “hypercoplanar” and so

(n−2)

· γ ∧ γ˙ ∧ . . . ∧ γ V

=0

⇔

˙ · =X φ(X)

(n) ... ¨ ∧ X ∧ ... ∧ X X

=0

Thus, identity between the slow manifold equation (11.3) given by the Generalized Tangent Linear System Approximation and the flow curvature manifold (11.2) is established. So, the Flow Curvature Method directly provides the slow manifold analytical equation associated with any n-dimensional slow-fast dynamical system. Then, invariance of the slow manifold will be established by using a concept introduced by Gaston Darboux (Cf. Darboux invariance theorem, Ch. 6).

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11.2

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187

Invariance

According to Schlomiuk (1993a,b) and LLibre and Medrado (2007) it seems that in his memoir entitled: Sur les équations différentielles algébriques du premier ordre et du premier degré, Gaston Darboux (1878, p. 71, 1878c, p.1012) has been the first to define the concept of invariant manifold. Let’s state the following proposition. = 0 where φ is a Proposition 11.2. The slow manifold defined by φ(X) 1 C in an open set U is invariant with respect to the flow of (11.1) if there and called cofactor which satisfies: exists a C 1 function denoted k(X) = k(X)φ( X) LV φ(X) ∈ U and where for all X Proof.

3

· ∇φ = LV φ = V

(11.5)

n ∂φ $ dφ . x˙ i = dt i=1 ∂xi

Lie derivative of the flow curvature manifold (11.2) reads:

˙ · =X LV φ(X)

(n+1) (n+1) ... ... ¨ ¨ ˙ ..., X ∧... ∧ X X, ∧X X, X = X,

(11.6)

¨ = J X ˙ where J is the functional jacobian Moreover, from the identity X matrix associated with any n-dimensional slow-fast dynamical system (11.1) it can be established that: (n+1)

= J nX ˙ X

if

dJ =0 dt

(11.7)

¨ = J X ˙ ⇔ γ = J V . where J n represents the nth power of J, e.g., X Then, it follows that (n+1)

(n)

= JJ n−1 X ˙ = J X X

3 See


(11.8)

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(n+1)

Replacing X

in Eq. (11.6) with Eq. (11.8) we have:

=X ˙ · LV φ(X)

(n) ... ∧ ...∧ J X ¨ ∧ X X

(n) ... ...,J X ˙ X, ¨ X, = X,

(11.9)

Then, identity (A.16) established in appendix leads to: = T r (J) X ˙ · LV φ(X)

(n) ... ¨ X ∧ X ∧... ∧ X

= k(X)φ( = T r (J) φ(X) X)

= T r (J) represents the trace of the functional jacobian matrix. where k(X) So, according to Prop. 11.2 invariance of the slow manifold analytical equation of any n-dimensional slow-fast dynamical system is established provided that the functional jacobian matrix is locally stationary (Cf. Eq. 11.7). Remark. Let’s recall that, according to Th. 8.1, all fixed points of the ndimensional slow-fast dynamical system (11.1) belong to the slow invariant manifold analytical equation (11.2).

11.3

Flow Curvature Method – Singular Perturbation Method

In this section a comparison between the Geometric Singular Perturbation Theory (Fenichel, 1979) recalled in Ch. 5 and Flow Curvature Method (Cf. Ch. 7) will highlight that, since this latter uses neither eigenvectors nor asymptotic expansions but simply involves time derivatives of the velocity vector field, curvature of the flow constitutes a general method simplifying and improving the slow invariant manifold analytical equation determination of any high-dimensional slow-fast dynamical systems. Let’s consider a n-dimensional singularly perturbed system (Cf. Ch. 5)

εx˙ = f (x, z, ε) z˙ = g (x, z, ε)

(11.10)

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The flow curvature manifold, i.e. the slow invariant manifold (11.2) associated with system (11.10) may be written as:

φ (x, z, ε) = 0

(11.11)

plug the perturbation expansion: X (z, ε) = X0 (z ) + εX1 (z ) + Let’s 2 O ε into Eq. (11.11) and solve order by order for X (z , ε) the Taylor z , ε), z , ε) up to terms of suitable order in ε: series expansion for φ(X( z , ε = φ X 0 , z, 0 + εD x φ X 0 , z , 0 0 , z, 0 X 1 + ε ∂φ X φ X, ∂ε

(11.12)

Order ε0 provides: 0 (z ), z, 0) = 0 φ(X

(11.13)

0 (z) due to the invertibility of D x f and the Implicit Funcwhich defines X tion Theorem. Order ε1 provides: 0 (z ) , z, 0 = 0 0 (z) , z, 0 X 1 (z) + ∂φ X D x φ X ∂ε

(11.14)

1 (z). which yields X 0 (z ) , z, 0) Applying the chain rule to the functional jacobian of φ(X according to the variable z and to the variable ε Eq. (11.14) may be written: 0g X 0 , z, 0 1 = D z X 0 , z, 0 − ∂ f X 0 , z, 0 X D x f X ∂ε

(11.15)

A comparison between Eqs. (11.13) and (5.5), Eqs. (11.15) and (5.6) highlights identity of both Flow Curvature Method and Geometric Singular Perturbation Theory up to suitable orders4 in ε. 4 Cf.

supra

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11.3.1

Darboux invariance – Fenichel’s invariance

According to Fenichel’s persistence theorem (Cf. Th. 5.1) a slow invariant (z, ε) the manifold Mε may be written as an explicit function: x = X invariance of which implies that X (z , ε) satisfies: (z, ε) g X (z, ε) , z, ε = f X (z, ε) , z, ε εD z X

(11.16)

Let’s write the slow manifold Mε as an implicit function by posing:

(z , ε) φ(x, z , ε) = x − X

(11.17)

According to Darboux invariance theorem (Cf. infra) Mε is invariant if its Lie derivative reads:

LV φ(x, z , ε) = k(x, z, ε)φ(x, z, ε)

(11.18)

Plugging Eq. (11.17) into the Lie derivative (11.18) leads to:

(z, ε) z˙ = k(x, z, ε)φ(x, z, ε) LV φ(x, z , ε) = x˙ − D x X which may be written according to Eq. (11.10):

LV φ(x, z, ε) =

1 (z, ε) g (x, z, ε) = k(x, z, ε)φ(x, z , ε) f (x, z, ε) − εD x X ε

Evaluating this Lie derivative in the location of the points where (z, ε) leads to: φ(x, z, ε) = 0, i.e. x = X (z , ε) g (X (z, ε) , z, ε) = 0 (z, ε) , z, ε) − εD x X (z , ε) , z, ε) = 1 f(X LV φ(X ε which is exactly identical to Eq. (11.16) used by Fenichel.

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11.3.2

191

Slow invariant manifold

In this subsection a link between the slow invariant manifold associated with any n-dimensional dynamical system singularly perturbed given by the Geometric Singular Perturbation Theory and that obtained by the Flow Curvature Method will be established. Proposition 11.3. The slow invariant manifold associated with any ndimensional dynamical system singularly perturbed may be deduced from its flow curvature manifold up to suitable orders in ε. Proof. The proof is still based on the concept of invariant manifolds (Cf. Ch. 9) and involves the method developed in Ch. 10. Two-dimensional dynamical systems According to Th. 7.3 and Def. 7.2 the flow curvature manifold of a two-dimensional dynamical system singularly perturbed (11.10) reads: + + + ˙ ¨ + + +X ∧ X κ1 = + + 3 + ˙ + +X +

⇔

+ + ¨ + ˙ ∧ X ε) = + φ(X, +=0 +X

⇔

φ (x, y, ε) = 0

Writing the total differential of the 1st flow curvature manifold leads to: dφ (x, y, ε) =

∂φ ∂φ ∂φ dx + dy + dε = 0 ∂x ∂y ∂ε

(11.19)

According to the Geometric Singular Perturbation Theory (Ch. 5, Sec. 5.2) the slow invariant manifold may be written as: y = Y (x, ε) = Y0 (x) + εY1 (x) + ε2 Y2 (x) + ε3 Y3 (x) + O ε4

(11.20)

∂Y ∂Y Replacing in Eq. (11.19) dy by its total differential dy = dx + dε ∂x ∂ε yields dφ(x, y, ε) =

∂φ ∂φ ∂Y + ∂x ∂y ∂x

dx +

∂φ ∂φ ∂Y + ∂ε ∂y ∂ε

dε

(11.21)

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According to Def. 9.1 φ(x, y, ε) is globally invariant iff dφ(x, y, ε) = 0, i.e. if  ∂φ ∂φ ∂Y   + =0    ∂x ∂y ∂x  ∂φ ∂φ ∂Y     ∂ε + ∂y ∂ε = 0

∂φ

⇔

∂Y ∂x = − ∂φ ∂x ∂y

⇔

∂Y ∂ε = − ∂φ ∂ε ∂y

∂φ

(11.22)

By replacing y = Y (x, ε) by its expression (11.20) in both parts of Eq. (11.22) and by setting

a10 = −

∂φ ∂x ∂φ ∂y y=Y (x,ε)

and

a01 = −

∂φ ∂ε ∂φ ∂y y=Y (x,ε)

leads to: = a10 y=Y (x,ε) ∂φ ∂ε = Y1 (x) + 2εY2 + O ε3 = − ∂φ = a01 ∂y

 (x,ε)   ∂Y∂x = Y0 (x) + εY1 + O ε2 = −    ∂Y (x,ε)    ∂ε

∂φ ∂x ∂φ ∂y

(11.23)

y=Y (x,ε)

By using a recurrence reasoning it may be stated that the functions Yi (x) of the slow invariant manifold associated with a two-dimensional singularly perturbed dynamical system: Y (x, ε) = Y0 (x) + εY1 (x) + ε2 Y2 (x) + ε3 Y3 (x) + O ε4 are given by the following expressions:  1 ∂ n a10   Y with n ≥ 0 (x) = lim  n ε→0 n! ∂εn  1 ∂ n−1 a01   Yn (x) = lim with n ≥ 1 ε→0 n! ∂εn−1

(11.24)

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MF 30

193

Example 11.1. Van der Pol model Let’s consider the model of Van der Pol (1926): 1 x+y− x ˙ f (x, y) V = = ε y˙ g (x, y) −x

x3 3

By using Geometric Singular Perturbation Theory (Cf. Ch. 5) the slow manifold equation associated with this system may written as a regular kbe perturbation expansion up to a suitable order O ε : y = Y (x, ε) = Y0 (x) + εY1 (x) + ε2 Y2 (x) + O ε3 where functions Yi (x) with i = 0, . . . , k have been computed in Ch. 5: 2 x x3 2x 1 + x −x+ε y = Y (x, ε) = +ε + O ε3 4 2 2 3 1−x (1 − x ) By using the Flow Curvature Method, i.e. according to Prop. 11.1 the slow manifold equation associated with this system reads: = φ (x, y, ε) = 9y 2 + 9x + 3x3 y + 6x4 − 2x6 + 9x2 ε = 0 φ(X) All functions Yi (x) may be deduced from this implicit equation by using the method developed above, i.e. according to Eqs. (11.24). Order ε0 : Y0 (x) = lim [a10 ] = −1 + x2 ε→0

from which one deduces that:

Y0 (x) =

x3 − x + C0 3

where the constant C0 may be chosen such that the singular approximation can be found again (C0 = 0).

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194

Order ε1 :  ∂a10 2x + (−1 + 3x2 )Y1 (x)   lim =  Y1 (x) = ε→0 ∂ε x − x3  x C1   Y1 (x) = + 1 − x2 x(−1 + x2 ) By choosing C1 = 0 one find again exactly the same functions Y1 (x) as those given by Geometric Singular Perturbation Theory, i.e. Y1 (x) =

x 1 − x2

Order ε2 : 2  1 ∂ a10 4(x + x3 ) − 2(−1 + x2 )3 (−1 + 3x2 )Y2 (x)   (x) = lim Y =  2 2 ε→0 2 ∂ε x(−1 + x2 )4  1 ∂a01 C2 x   Y2 (x) = lim + = ε→0 2 ∂ε (1 − x2 )3 x(−1 + x2 ) Then, by choosing C2 = 0 one find the function Y2 (x): Y2 (x) =

x (1 − x2 )3

which is different from that given by Geometric Singular Perturbation Theory: Y2 (x) =

x(1 + x2 ) (1 − x2 )4

Another method for transforming the slow invariant manifold implicit equation given by the Flow Curvature Method would have consisted in plugging in it the regular perturbation expansion (Cf. Ch. 5) and then identifying order by order. y = Y0 (x) + εY1 (x) + ε2 Y2 (x) + O ε3 Thus, in both cases the flow curvature manifold reads: y=

x x x3 −x+ε + ε2 + O ε3 3 2 2 3 1−x (1 − x )

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So, as expected by Professor E. Benoˆıt (personal communication), both slow manifolds are completely identical up to order one in ε. At order ε2 a difference appears which is due to the fact that the 1st flow curvature manifold associated with a two-dimensional dynamical system is defined by the second order tensor of curvature 5 , i.e. by a determinant involving the first and second time derivatives of the velocity vector field. If one makes the same computation as previously but with the Lie derivative of the flow curvature manifold which is defined by a determinant containing the first and third time derivatives of the velocity vector field (third order tensor ), then there is no more difference between order two in ε and functions Y2 (x) given by both methods are exactly the same. This result had been already found by Bruno Rossetto (Rossetto, 1986b) by using successive approximation method. Moreover, it may be established that since flow curvature manifold lies in the ε-vicinity of the singular approximation of any n-dimensional slow-fast dynamical system (Cf. Fig. 11.1) it can be used to define its slow manifold analytical equation. Y 2

1

0

X

1

2 2

Fig. 11.1

1

0

1

2

Van der Pol slow invariant manifold

In Fig. 11.1 Van der Pol slow manifold has been plotted according to Flow Curvature Method (in blue) and Singular Perturbation Method (in magenta). 5 1st

curvature

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Remark. For dimension greater than three, slow manifold determination with the Geometric Singular Perturbation Theory leads to tedious calculations while Flow Curvature Method directly provides the slow manifold analytical equation of any n-dimensional slow-fast dynamical system. Three-dimensional dynamical systems According to Th. 7.4 and Def. 7.2 the flow curvature manifold of a three-dimensional dynamical system singularly perturbed (11.10) reads: ... ¨ ∧ X ˙ · X X κ2 = + +2 + ˙ ¨ + + +X ∧ X

... ¨ ˙ ⇔ φ(X) = X · X ∧ X = 0

⇔ φ (x, y, z, ε) = 0

Writing the total differential of the 2nd flow curvature manifold leads to: dφ (x, y, z, ε) =

∂φ ∂φ ∂φ ∂φ dx + dy + dz + dε = 0 ∂x ∂y ∂z ∂ε

(11.25)

According to the Geometric Singular Perturbation Theory (Ch. 5, 5.2) the slow invariant manifold may be written as: z = Z (x, y, ε) = Z0 (x, y) + εZ1 (x, y) + ε2 Z2 (x, y) + O ε3 Replacing dz by its total differential dz =

(11.26)

∂Z ∂Z ∂Z dx + dy + dε yields ∂x ∂y ∂ε

in Eq. (11.25) dφ =

∂φ ∂φ ∂Z + ∂x ∂z ∂x

dx +

∂φ ∂φ ∂Z + ∂y ∂z ∂y

dy +

∂φ ∂φ ∂Z + ∂ε ∂z ∂ε

dε (11.27)

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According to Def. 9.1 φ(x, y, ε) is globally invariant iff dφ(x, y, ε) = 0, i.e. if  ∂φ ∂φ ∂Z   + =0    ∂x ∂z ∂x      ∂φ ∂φ ∂Z + =0  ∂y ∂z ∂y       ∂φ ∂φ ∂Z   + =0  ∂ε ∂z ∂ε

∂φ

⇔

∂Z ∂x = − ∂φ ∂x ∂z

⇔

∂Z ∂y = − ∂φ ∂y ∂z

⇔

∂Z ∂ε = − ∂φ ∂ε ∂z

∂φ

(11.28)

∂φ

By replacing z = Z (x, y, ε) by its expression (11.26) in both parts of Eq. (11.28) and by setting ∂φ ∂φ ∂φ ∂y ∂x ∂ε ; a010 = − ∂φ ; a001 = − ∂φ a100 = − ∂φ ∂z z=Z(x,y,ε)

∂z z=Z(x,y,ε)

∂z z=Z(x,y,ε)

leads to:  2 ∂Z(x,y,ε) ∂Z1 0  =− = ∂Z  ∂x ∂x + ε ∂x + O ε     2 ∂Z(x,y,ε) ∂Z1 0 =− = ∂Z ∂y ∂y + ε ∂y + O ε       ∂Z(x,y,ε) = Z1 + 2εZ2 + O ε3 = − ∂ε

= a100 z=Z(x,y,ε) = a010 z=Z(x,y,ε) ∂φ ∂ε = a001 ∂φ

∂φ ∂x ∂φ ∂z ∂φ ∂y ∂φ ∂z

∂z

(11.29)

z=Z(x,y,ε)

By using a recurrence reasoning it may be stated that the functions Zi (x) of the slow invariant manifold associated with a three-dimensional singularly perturbed dynamical system: z = Z (x, y, ε) = Z0 (x, y) + εZ1 (x, y) + ε2 Z2 (x, y) + O ε3 are given by the following expressions:  ∂Zn (x, y) 1 ∂ n a100   = lim with n ≥ 0   ε→0 n! ∂εn ∂x      1 ∂ n a010 ∂Zn (x, y) = lim with n ≥ 0 ε→0 n! ∂εn  ∂y       1 ∂ n−1 a001   Zn (x) = lim with n ≥ 1 ε→0 n! ∂εn−1

(11.30)

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MF 31

Example 11.2. Chua’s system Now, let’s consider the system of Chua et al. (1986):      1 3 2 x˙ f (x, y, z) ε z − c1 x − c2 x − µx  g (x, y, z)  =   y˙  =  V −z −ax + y + bz z˙ h (x, y, z) By using Geometric Singular Perturbation Theory (Cf. Ch. 5) the slow manifold equation associated with this system may written as a regular be perturbation expansion up to a suitable order O εk : z = Z (x, y, ε) = Z0 (x, y) + εZ1 (x, y) + ε2 Z2 (x, y) + O ε3 where the functions Zi (x) with i = 1, 2 have been computed in Ch. 5: z = c1 x3 + c2 x2 + µx + ε

−ax + y + bc1 x3 + bc2 x2 + bxµ + O ε2 2 3c1 x + 2c2 x + µ

By using the Flow Curvature Method, i.e. according to Prop. 11.1 the slow manifold equation associated with this system leads to an algebraic polynomial6 of degree 9, 3, 3 in x, y, z respectively: φ(x, y, z, ε) = 0 (Cf. Mathematica File). All functions Zi (x) may be deduced from this implicit equation by using the method developed above, i.e. according to Eqs. (11.30). Order ε0 :  ∂Z0 (x, y)   = lim [a100 ] = 3c1 x2 + 2c2 x + µ  ε→0 ∂x  ∂Z0 (x, y)   = lim [a010 ] = 0 ε→0 ∂y from which one deduces that: Z0 (x, y) = Z0 (x) = c1 x3 + c2 x2 + µx + C0 where the constant C0 may be chosen such that the singular approximation can be found again (C0 = 0). 6 Cf.

Mathematica file for flow curvature manifold equation

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199

Order ε1 : Z1 (x, y) = lim [a001 ] = ε→0

−ax + y + bc1 x3 + c2 bx2 + bxµ 3c1 x2 + 2c2 x + µ

Then, one finds again exactly the same functions Z0 (x) and Z1 (x, y) as those given by Geometric Singular Perturbation Theory for orders ε0 , ε1 and also ε2 (Z2 (x, y)). Another method for transforming the slow invariant manifold implicit equation given by the Flow Curvature Method would have consisted in plugging in it the regular perturbation expansion (Cf. Ch. 5) and then identifying order by order. z = Z0 (x) + εZ1 (x) + ε2 Z2 (x) + O ε3 Then, the flow curvature manifold reads: z = c1 x3 + c2 x2 + µx + ε

−ax + y + bc1 x3 + bc2 x2 + bxµ + O ε2 3c1 x2 + 2c2 x + µ

1.0

0.5

0.0

0.5

1.0 1.0

Fig. 11.2

0.8

0.6

0.4

0.2

0.0

Chua’s slow invariant manifold in (xz)-plane

In Fig. 11.2 Chua’s slow manifold is plotted according to Flow Curvature Method (in blue) and Singular Perturbation Method (in magenta).

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Here again, as expected by Professor E. Benoˆıt, both slow manifolds are completely identical up to order two in ε. At order ε3 a difference appears which is due to the fact that the 2nd flow curvature manifold associated with a three-dimensional dynamical system is defined by the third order tensor of torsion 7 , i.e. by a determinant involving the first, second and third time derivatives of the velocity vector field. If one makes the same computation as previously but with the Lie derivative of the flow curvature manifold which is defined by a determinant containing the first, second and fourth time derivatives of the velocity vector field (third order tensor ), then there is no more difference between order three in ε and functions Z3 (x, y) given by both methods are exactly the same. 11.4

Non-singularly perturbed systems

Then main difference between Geometric Singular Perturbation Theory and Flow Curvature Method is that the former which is based on regular expansions including a small multiplicative parameter deals only with lowdimensional two and three singularly perturbed systems while the latter can be used for any kind of n-dimensional dynamical system singularly perturbed or not since it does not require any regular expansions since it only involves time derivatives of the velocity vector field. So, as an example let’s focus on the model of Lorenz (1963) which is not singularly perturbed but considered as a slow fast dynamical system 8 and for which the Geometric Singular Perturbation Theory fails to provide its slow manifold.       x˙ f (x, y, z) σ(y − x)  g (x, y, z)  =  −xz + rx − y  V  y˙  = z˙ h (x, y, z) xy − bz with the following set of parameters σ = 10, b =

7 2nd 8 Cf.

curvature Ch. 5, 5.3.2

8 and by setting r = 28. 3

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MF 32

201

According flow curvature manifold equation associated with this system reads: φ(x, y, z) = −12954816x4 + 21168x6 + 13331304x3y − 2772x5 y − 15012x4 y 2 + 27x6 y 2 − 6554520xy 3 + 13410x3y 3 + 340200y 4 + 15120x2 y 4 − 8100xy 5 + 15906240x2z + 1311744x4z − 1512x6 z + 5112720xyz − 1454430x3yz − 45x5 yz − 5680800y 2z − 720x2 z 4 − 790440x2y 2 z + 540x4 y 2 z + 456180xy 3z − 810x3 y 3 z + 1800y 4z − 1686544x2z 2 − 45252x4z 2 + 27x6 z 2 − 317920xyz 2 + 58320x3yz 2 + 372800y 2z 2 + 15750x2 y 2 z 2 − 8100xy 3 z 2 + 59040x2z 3 + 540x4 z 3 + 9297666x2y 2 + 6480xyz 3 − 810x3 yz 3 − 7200y 2z 3 = 0

(11.31)

Remark. Let’s note that flow curvature manifold exhibits the symmetry of Lorenz model, i.e. φ(−x, −y, z) = φ(x, y, z). Such symmetry will be used in Ch. 13 (Cf. supra).

Fig. 11.3

Lorenz slow invariant manifold

In Fig. 11.3 the slow manifold analytical equation (11.19) associated with Lorenz model has been plotted according to Flow Curvature Method.

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Chapter 12

Integrability

“If one looks at the different problems of the integral calculus which arise naturally when one wishes to go deep into the different parts of physics, it is impossible not to be struck by the analogies existing. Whether it be electrostatics or electrodynamics, the propogation of heat, optics, elasticity, or hydrodynamics, we are led always to differential equations of the same family.” — H. Poincaré —

In the context of dynamical systems the notion of integrability refers to the existence of manifolds invariant with respect to the flow of such systems. This concept has a refinement in the case of Hamiltonian systems, known as complete integrable in the sense of Liouville (1839) (Cf. Ch. 2). As pointed out in Chap. 6 existence and number of invariant manifolds for a dynamical system is of great importance since it enables, according to Darboux theory of integrability, to build first integrals of such system. 12.1

First integral

As for invariant sets, it can be considered that first integrals may be either local or global. 12.1.1

Global first integral

= 0 of dynamical Definition 12.1. A global first integral manifold φ(X) system (2.1) is defined as: =0 LV φ(X)

∈ Rn ∀X

203

for all

t∈R

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Example 12.1. Spherical pendulum Let’s consider the dynamical system of Ex. 2.13: 

     x˙1 f1 (x1 , x2 , x3 , x4 ) −ω1 x2  x˙2       f2 (x1 , x2 , x3 , x4 )  =  ω1 x1  = V   x˙3   f3 (x1 , x2 , x3 , x4 )   −ω2 x4  x˙4

f4 (x1 , x2 , x3 , x4 )

ω2 x3

Flow curvature manifold 1 associated with this dynamical system reads: = φ(x1 , x2 , x3 , x4 ) = (x2 + x2 )(x2 + x2 ) φ(X) 1 2 3 4 is identically zero it confirms, according Since the Lie derivative of φ(X) to Prop. 9.2, that each factor of the flow curvature manifold is first integral of this dynamical system. 12.1.2

Local first integral

= 0 be a manifold lying locally in the vicinity of Definition 12.2. Let ψ(X) = 0. A local first integral manifold φ(X) = 0 of dynamical a manifold φ(X) system (2.1) is defined as: =0 LV φ(X) MF 33

∈φ ∀X

and

∈ψ X

Example 12.2. Van der Pol system Let’s consider the system of Van der Pol (1926): V

1 x+y− x˙ f (x, y) = = ε y˙ g (x, y) −x

x3 3

This singularly perturbed system has a singular approximation namely: =y+x− ψ(X) 1 The

3rd flow curvature manifold

x3 =0 3

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205

According to the Flow Curvature Method (Prop. 11.1) the slow manifold analytical equation associated with this system is: = 9y 2 + 9x + 3x3 y + 6x4 − 2x6 + 9x2 ε = 0 φ(X) Its Lie derivative may be written as: 2 3 = T r(J)φ(X) + 18x (x + y − x )2 LV φ(X) ε 3

x3 the Lie derivative reads: = ψ(X), 3 2 2 = T r(J)φ(X) + 18x ψ(X) LV φ(X) ε

Taking into account that y + x −

Y

2

1

0

X

1

2

2

Fig. 12.1

1

0

1

2

Local first integral of Van der Pol model

So, in Fig. 12.1 as soon as both slow manifold (in blue) and singular = approximation (in green) lie in the same vicinity, i.e. as soon as φ(X) 0 and ψ(X) = 0, LV φ(X) = 0. But, since according to the Geometric Singular Perturbation Theory (Cf. Ch. 5), both singular approximation and slow manifold are supposed to be in the same ε-vicinity, the slow manifold = 0) may be considered as a local first integral of Van der Pol model. (φ(X)

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206

12.2

Linear invariant manifolds as first integral

In Ch. 9 it has been established according to Prop. 9.1 that flow curvature manifold enabled to find linear invariant manifolds. In this section, it will be stated that such manifolds may be considered as algebraic general integrals, i.e. as first integrals. Proposition 12.1. Any n-dimensional dynamical system for which the curvature of the trajectory curve is identically zero admits a linear invariant manifold as a first integral. Proof.

n $

= a0 + a1 x1 + a2 x2 + . . . + an xn = a0 + Let ϕ(X)

ai xi

i=1

be a n-dimensional linear invariant manifold. So, according to Def. 9.1:

= k1 (X)ϕ( = a1 x˙ 1 + a2 x˙ 2 + . . . + an x˙ n = X) LV ϕ(X)

n

ai x˙ i

(12.1)

i=1

˙ X) = k2 (X), time-derivative of (12.1) leads to: Setting: (k 2 + k)( n ˙ X)ϕ( = (k 2 + k)( = k2 (X)ϕ( = LV LV ϕ(X) X) X) ai x ï i=1

Then, applying the Leibniz formulae the nth time derivative reads:

= LnV φ(X)

n−1 p=0

Cnp

p dn−p−1 d φ(X) k(X) n−p−1 dt dtp

= = kn (X)φ( X)

n

(n)

ai xi

(12.2)

i=1

The proof is quite similar to those of Prop. 9.1 Taking into account determinant properties (P1 ) and (P2 ) flow curvature manifold (7.18) reads: 

 x˙ 2 · · · k1 (X)  x ¨2 · · · k2 (X)   .. ..  . ··· .  (n) (n) x1 x2 · · · kn (X)

x˙ 1 x  ¨1 . = ϕ(X)det φ(X)  .. 

(12.3)

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207

Thus, the condition for which this determinant vanishes is another determinant property which states that if all the elements of a row (resp. a column) are null the determinant is null. By supposing that all the other rows (resp. columns) are not null nor identical the condition for which the flow curvature manifold vanishes is = 0. But, since all the other ki (X) with i > 1 consist of k1 or its k1 (X) is first integral. time derivative they all vanish too and so ϕ(X) Corollary 12.1. The linear invariant manifold, first integral of any ndimensional dynamical system for which the curvature of the trajectory curve is identically zero, is in factor of the osculating plane associated with this system. Proof.

Let suppose that the n-dimensional linear invariant manifold : = a0 + a1 x1 + a2 x2 + . . . + an xn = a0 + ϕ(X)

n

ai xi

i=1

is first integral of a n-dimensional dynamical system. So, according to Def. 12.1: = k1 (X)ϕ( = a1 x˙ 1 + a2 x˙ 2 + . . . + an x˙ n = X) LV ϕ(X)

n

ai x˙ i = 0 (12.4)

i=1

and time-derivatives of (12.4) leads to: = LnV φ(X)

n

(n)

ai xi = 0

(12.5)

i=1

The osculating plane (7.10) reads: 

x˙ 1 x˙ 2 x  ¨1 x¨2 = det  . .. Π(X)  .. .  (n−1) (n−1) x1 x2

· · · x˙ n ··· x ¨n .. ··· .

(n−1)

     

(12.6)

· · · xn

The proof is quite similar to those of Prop. 12.1. Taking into account is first determinant properties (P1 ) and (P2 ) and considering that ϕ(X) integral the osculating plane (12.6) reads:

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

x˙ 1 x  ¨1 = det  . Π(X)  .. 

 x˙ 2 · · · ϕ(X) x¨2 · · · 0   .. ..  . ··· .  

(n) (n) x1 x2

···

(12.7)

0

By using the Laplace expansion along the first row and last column for the computation of this determinant leads to: n+1 = ϕ(X)(−1) Π(X) |M1n |

where M1n is the minor matrix that results from deleting the ith row and the j th column of (12.7) MF 34

Example 12.3. First integral of Volterra-Lotka system Let’s consider the dynamical system (Darboux, 1878, p. 136) of Ex. 6.12 with the following set of parameters: b = b = b = 1, c = c = c = −1       x˙ f (x, y, z) x(y − z)  g (x, y, z)  =  y(−x + z)  V  y˙  = z˙ h (x, y, z) z(x − y) verifying Darboux condition of integrability (6.25): bb b + cc c = 0 It may be stated that its flow curvature manifold is identically zero, i.e. = 0. Thus, according to Converse proposition 9.1 and to Prop. 12.1 φ(X) this system admits a linear invariant manifold as first integral. Osculating plane equation passing through the origin reads: = xyz(x + y + z)Q(X) Π(X) is a polynomial of degree higher than one. Thus, it may be where Q(X) checked that x + y + z is first integral of this dynamical system. Moreover, according to Darboux theorem 6. (Cf. infra), α, β, γ and δ may be chosen such that a first integral of this system has the form: = ϕ(X)

(x + y + z)3 =C xyz

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Fig. 12.2

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209

Volterra-Lotka’s first integral

= C1 and x + y + z = C2 forming In Fig. 12.2 both first integrals ϕ(X) the general integral of Volterra-Lotka have been plotted. 12.3 12.3.1

Darboux theory of integrability General integral – Multiplier

Propositions 9.1 and 12.1 may be applied to the case of polynomial dynamical systems (6.12) in connection with Darboux theory of integrability. Thus, Darboux theorems 1 & 2 may be written as: Theorem 12.1. Let’s suppose that the flow curvature manifold associated with a polynomial dynamical systems (6.12) has q linear invariant i = 1, 2, . . . , q, in factor with q = Mn + n − 1 and manifolds ui = 0, m+n−1 Mn = . Then there exists q numbers αi such that the genn q = uα1 uα2 · · · uα eral integral (first integral) of this system reads: φ(X) q . 1 2 Proof.

Cf. infra p. 98.

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Theorem 12.2. Let’s suppose that the flow curvature manifold associated with a polynomial dynamical systems (6.12) has p linear invariant manifolds ui = 0, i = 1, 2, . . . , p, in factor with p = q − 1. Then there exists p numbers p = uα1 uα2 · · · uα αi such that the multiplier of this system is µ(X) p . 1 2 Proof.

Cf. infra p. 100.

Remark. All Darboux theorems 3, 4, 5, 6, 7 & 8 may be recast by considering that Darboux algebraic particular integrals are linear invariant manifolds or decomposable quadrics. MF 35

Example 12.4. First integral of quadratic system Consider the following quadratic system (Corbera and Llibre, 2006).       2 x˙ f (x, y, z) 1 + x2 − y 2   g (x, y, z)  =   y˙  = V xy   z˙ h (x, y, z) −xz The flow curvature manifold associated with this system reads: =0 = yz(x2 + 2(−1 + y)2 )(x2 + 2(1 + y)2 )Q(X) φ(X) is a polynomial of degree greater than one. Then, it can be where Q(X) checked that each factor is an invariant manifold. So, according to Th. 12.1 αi may be chosen such that a first integral of this system is: = y α1 z α2 (x2 + 2(−1 + y)2 ) α3 (x2 + 2(1 + y)2 ) α4 = C ϕ(X) with α1 − α2 + α3 + α4 = 0. By setting α1 = α4 = 1, α2 = 2 and α3 = 0 the first integral of this system has the form: = yz 2 (x2 + 2(1 + y)2 ) = C ϕ(X)

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Fig. 12.3

12.3.2

211

First integral of quadratic system

Darboux homogeneous polynomial dynamical systems of degree two

Since in the case of homogeneous polynomial dynamical systems of degree two Darboux approach consists in using linear invariant manifolds which may be deduced from the flow curvature manifold Darboux theorem 9 reads: Theorem 12.3. (i) If p is a factor of the flow curvature manifold and if pα (α = −4) is a multiplier of system (6.20) then a first integral of this system is ν = Cp3 with C an arbitrary constant.

(ii) If p and q are factors of the flow curvature manifold and if pm q m is a multiplier of system (6.20) then a first integral of this system is uα pβ pγ = C with C an arbitrary constant.

(iii) If p, q and r are factors of the flow curvature manifold and if pm q m rm is a multiplier of system multipliers of system (6.20) then a first integral of this system is pα q β rγ sδ = C with C an arbitrary constant. Proof.

Cf. G. Darboux 1878, p. 132 and next (Cf. infra).

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212

Example 12.5. Homogeneous dynamical system (Cairo, 2000) Consider the homogeneous polynomial dynamical system of Ex. 6.10:       x˙ f (x, y, z) x(x + y + z)  g (x, y, z)  =  y(−2x − y − z)   y˙  = V z˙ h (x, y, z) z(x − 2y + z) The flow curvature manifold associated with this system reads: = xyz(3x + 2y − 4z)Q(X) =0 φ(X) is a polynomial of degree higher than one. Then, it may be where Q(X) checked on the one hand that x, y, and z in factor in the flow curvature manifold are linear invariant manifolds, according to Prop. 9.1, and on the other hand that xα y β z γ is a multiplier2 of this dynamical system. So, according to Th. 12.1 a first integral of this system is pα q β rγ sδ = C. Then, since 3x + 2y − 4z in factor in the flow curvature manifold is also a linear invariant manifold α, β, γ and δ may be chosen3 such that:

αK1 + βK2 + γK3 + δK4 = 0 αh1 + βh2 + γh3 + δh4 = 0

It leads to α = −δ/3, β = 0, γ = −2δ/3. So, by setting δ = 3 a first integral of this system has the form: = ϕ(X)

(3x + 2y − 4z)3 =C xz 2

12.3.3

Planar polynomial dynamical systems

By considering that Darboux algebraic particular integrals are linear invariant manifolds or decomposable quadrics which may be deduced from the flow curvature manifold Darboux theorem for planar polynomial dynamical systems as follows. 2 According 3 According

to Darboux theorem 6 and 9, cf. infra to Darboux theorem 5 and to Th 12.1, cf. infra

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213

Theorem 12.4 (Darboux Theorem for planar dynamical systems). Suppose that the flow curvature manifold associated with a polynomial dynamical system (6.34) of degree m admits q linear invariant manifolds fi with cofactors Ki , for i = 1, . . . , q. λ

+ 1, then the function f1λ1 · · · fq q , for suitable λi ∈ R2 q $ not all zero, is a first integral and λi Ki = 0. m(m+1) 2

(a) If q

i=1 λ

+ 1, then the function f1λ1 · · · fq q , for suitable λi ∈ R2 q $ not all zero, is a first integral and λi Ki = 0, or an integrating factor m(m+1) 2

(b) If q =

and

q $

i=1

λi Ki = −Div( ).

i=1

(c) If q < m(m+1) and there exists λi ∈ R2 not all zero such that 2 q $ λ λi Ki = 0, then f1λ1 · · · fq q is a first integral. i=1

(d) If q
2. This implies to distinguish the two-dimensional case from the others. Then, starting from this conjectures and taking into account the Pierre Curie’s principle it is possible to deduce the following important results about the components of the velocity vector field or phase space of any n-dimensional dynamical system.

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13.2.1

DGeometry

219

Two-dimensional polynomial dynamical systems

The case of two-dimensional dynamical systems is very special since a relation between the parity of flow curvature manifold and that of the velocity vector field may be established. Let’s consider the two-dimensional dynamical system (13.1) and let’s suppose that its flow curvature manifold exhibits a parity, i.e. is either symmetric or anti-symmetric. Then, the following theorem may be stated. Theorem 13.1. If the flow curvature manifold associated with a twodimensional dynamical system is such that: X) X) = ±φ(X) then (− = ∓ ( φ(−X) Proof. The proof is based on the derivative of symmetric functions, e.g., the derivative of an odd function is an even function and vice versa. As a consequence time derivative of a symmetric velocity vector field, i.e. acceleration vector field is necessarily an anti-symmetric function as proved below. Let’s consider a n-dimensional velocity vector field: X) X) X) (X) = ( with the following symmetry: (− = ( V According to Eq. (7.5) acceleration vector field may be written: ˙ X) = J(X) V (X) = ( γ (X) which consists entirely of partial The functional Jacobian matrix J(X) derivatives of the velocity vector field with respect to coordinates is anti = −J(X). symmetric to the velocity vector field, i.e. J(−X) Thus, acceleration vector field is an anti-symmetric function1 , i.e. = −γ (X) γ (−X)

⇐⇒

˙ X) ˙ X) = − ( (−

As a consequence, whatever the symmetry of the velocity vector field, acceleration vector field is necessarily an anti-symmetric function. So, the flow curvature manifold associated with a two-dimensional vector field which is deduced from the wedge product between velocity and acceleration vector field is anti-symmetric (resp. symmetric) if the velocity vector field symmetric (resp. anti-symmetric). 1 An

anti-symmetric velocity vector field provides the same result.

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Example 13.3. Van der Pol model It may be checked that flow curvature manifold associated with Van der Pol model (Cf. Ex. 13.1, Eq. (13.1)) exhibits the following parity: φ(−x, −y) = φ(x, y). Then, according to Th. 13.1 the velocity vector field is anti-symmetric, i.e. (−x, −y) = − (x, y). Starting from dimension three flow curvature manifold is still either symmetric or anti-symmetric. Then, no conclusion can be drawn concerning the vector field which may be either symmetric or anti-symmetric too. But, then parity of the flow curvature manifold may be related to that of the phase space. 13.2.2

n-dimensional polynomial dynamical systems

Let’s consider a n-dimensional dynamical system and let’s suppose that its flow curvature manifold exhibits a parity, i.e. is either symmetric or anti-symmetric. Then, the following proposition may be stated. Theorem 13.2. X) exhibit the • All time derivatives of even order of velocity vector field ( X), i.e. same symmetry as the velocity vector field ( (2k)

(2k)

(−X) (X) X) =± = (− X) exhibit the • All time derivatives of odd order of velocity vector field ( → −X, i.e. anti-symmetry X (2k+1)

(−X) =−

(2k+1)

(X)

where k is a positive integer. Proof.

The proof is still based on the derivative of symmetric functions.

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Corollary 13.3. The flow curvature manifold parity is that of the phase space, i.e.

= (−1)n φ(X) φ(−X) where n > 2 is the dimension of the phase space. Proof. Proof may be deduced from the previous Th. 13.2. According to Def. 7.2 flow curvature manifold consists of the determinant of the time X) which are alternatively of even derivatives of the velocity vector field ( and odd order. As an example, flow curvature manifold associated with a three-dimensional dynamical system is still anti-symmetric. In such cases velocity vector field may be either symmetric or anti-symmetric and no conclusion can be drawn. Example 13.4. Flow curvature manifold parity Let’s consider the following polynomial dynamical system:      n x˙ f (x, y, z) x  g (x, y, z)  =  y n   y˙  = V zn z˙ h (x, y, z) It may be checked that the flow curvature manifold associated with this system is still antisymmetric whatever the exponent n is odd or even, i.e. whatever the velocity vector field is symmetric or antisymmetric. Remark. Starting from dimension three, dynamical systems admit more generally a symmetry than a parity which is a “strong” property nevertheless it should be of great interest to deeply study the symmetry of the flow curvature manifold in many cases in order to classify the various possibilities of symmetry of vector fields. Thus, according to these results it is possible to solve (in some cases) the inverse problem. Some classical examples are presented in the next section.

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13.3

Inverse problem for polynomial dynamical systems

13.3.1

Two-dimensional polynomial dynamical systems

Let’s suppose that the only knowledge about a dynamical system is its flow curvature manifold: = 9y 2 + 9x + 3x3 y + 6x4 − 2x6 + 9x2 ε = 0 φ(X)

(13.4)

which is a polynomial of maximum degree 6 (in x) with the symmetry: φ(−x, −y) = φ(x, y) According to Prop. 13.1 d = max (n + 2m − 1, 2n + m − 1) = 6, then either n + 2m − 1 = 6 or 2n + m − 1 = 6. This leads to (n, m) = (3, 2) or (n, m) = (3, 1). It may be checked that the former couple will provide a polynomial of maximum degree 7, so let’s consider the latter and pose:  2 3 $ $   aij xi y j  f (x, y) = i=0 j=0

1 $ 1 $   bij xi y j  g (x, y) =

(13.5)

i=0 j=0

with aij = 0 if i + j > 3 and bij = 0 if i + j > 1. According to Corollary 8.1 the flow curvature manifold (13.4) admits the origin for fixed point. Theorem 13.1 implies that the components of the vector field of the dynamical system are such that: f (−x, −y) = −f (x, y) and g (−x, −y) = −g (x, y) so,

a0 = b0 = a20 = a11 = a02 = 0

(13.6)

The vector field may thus be written:

f (x, y) = a10 x + a01 y + a30 x3 + a21 x2 y + a12 xy 2 + a30 y 3 g (x, y) = b10 x + b01 y

(13.7)

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Inverse Problem

223

Then, by computing the flow curvature manifold associated with vector field (13.7) and comparing terms to terms with the “original” flow curvature manifold (13.4) leads to a system of nonlinear equations which may be solved (numerically) by using a Groebner basis. The result is: a03 = a12 = a21 = b01 = 0, b10 = −1, a01 = a10 =

1 1 , a30 = − (13.8) ε 3ε

By replacing in Eq. (13.7), the vector field reads:

f (x, y) = 1ε x + 1ε y − g (x, y) = −x

1 3 3ε x

and corresponds to the Van der Pol model. 13.3.2

Three-dimensional polynomial dynamical systems

Let’s suppose that the only knowledge about a dynamical system is its flow curvature manifold φ(x, y, z) = −12954816x4 + 21168x6 + 13331304x3y − 2772x5 y − 15012x4 y 2 + 27x6 y 2 − 6554520xy 3 + 13410x3y 3 + 340200y 4 + 15120x2 y 4 − 8100xy 5 + 15906240x2z + 1311744x4z − 1512x6 z + 5112720xyz − 1454430x3yz − 45x5 yz − 5680800y 2z − 720x2 z 4 − 790440x2y 2 z + 540x4 y 2 z + 456180xy 3z − 810x3 y 3 z + 1800y 4z − 1686544x2z 2 − 45252x4z 2 + 27x6 z 2 − 317920xyz 2 + 58320x3yz 2 + 372800y 2z 2 + 15750x2 y 2 z 2 − 8100xy 3 z 2 + 59040x2z 3 + 540x4 z 3 + 9297666x2y 2 + 6480xyz 3 − 810x3 yz 3 − 7200y 2z 3 = 0

(13.9)

which is a polynomial of maximum degree 8 (in x6 y 2 ). According to Prop. 13.2 this leads to many cases for which it may be checked that they do not provide a polynomial of degree 8 and then the couple “solution” is (n, m, l) = (1, 2, 2).

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So let’s pose:  1 $ 1 $ 1 $   f (x, y, z) = aijk xi y j z k    i=0 j=0 k=0    2 $ 2 $ 2 $ g (x, y, z) = bijk xi y j z k  i=0 j=0 k=0    2 $ 2 $ 2 $    cijk xi y j z k  h (x, y, z) =

(13.10)

i=0 j=0 k=0

with aijk = 0 if i + j + k > 1, bijk = 0 and cijk = 0 if i + j + k > 2. According to Corollary 8.1 the flow curvature manifold (13.8) admits the origin for fixed point. Moreover, due to the rotation symmetry φ (−x, −y, z) = φ (x, y, z) the components of the vector field of the dynamical system are such that:   f (x, y, z) = a100 x + a010 y g (x, y, z) = b100 x + b010 y + b101 xz  h (x, y, z) = c001 z + c110 xy

(13.11)

Then, by computing the flow curvature manifold associated with vector field (13.11) and comparing terms to terms with the “original” flow curvature manifold (13.9) leads to a set of nonlinear equations which may be solved (numerically) by using a Groebner basis. The result is: a100 = a010 = σ, b100 = r, b010 = b101 = −1, c001 = −b, c110 = 1 (13.12) By replacing in Eq. (13.12), the vector field reads:   f (x, y, z) = σ (y − x) g (x, y, z) = rx − y − xz  h (x, y, z) = −bz + xy and corresponds to the Lorenz model. Remark. According to the dimension of the dynamical system and to the solvability of the nonlinear system such results lead to a family of vector fields and not always exactly to the original dynamical system.

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PART 3

Applications

The mind uses its faculty for creativity only when experience forces it to do so. — H. Poincaré —

225

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226

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Chapter 14

Dynamical Systems

In this chapter, theorems established in Chapter 8 are applied to lowdimensional two and three dynamical systems in order to find their fixed points and to define their stability.

14.1 MF 38

FitzHugh-Nagumo model

The FitzHugh-Nagumo model (FitzHugh, 1961; Nagumo et al., 1962) is a simplified version of the Hodgkin-Huxley model which models in a detailed manner activation and deactivation dynamics of a spiking neuron. In the original papers of FitzHugh this model was called “Bonhoeffer-Van der Pol oscillator”, since it contains the Van der Pol oscillator as a special case for a = b = 0. V

1 x˙ f1 (x, y) = ε (x(1 − x)(x + a) − y + b) = f2 (x, y) x − cy y˙

(14.1)

x denotes the membrane potential, y is a recovery variable and b is the magnitude of stimulus current with parameter ε = 0.01, a = 0.1, b = 0.01 and c = 0.5 and where the functions fi are infinitely differentiable with respect to all x, y and t, i.e., are C ∞ functions in a compact E included in R2 and with values in R. According to Th. 8.1 and Corollary 8.1 it may be stated that the FitzHugh-Nagumo model admits1 one fixed point I(xI , yI ). Then, since the Hessian of the flow curvature manifold evaluated at the equilibrium I is positive, i.e., Hφ(X ∗ ) > 0 this fixed point is a focus according to Th. 8.3. 1 Cf.

Mathematica file 227

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Remark. By using the same theorems it can be shown that Brusselator’s model admits a focus as unique fixed point (Cf. Mathematica file). 14.2

MF 39

Pikovskii-Rabinovich-Trakhtengerts model

The Pikovskii-Rabinovich-Trakhtengerts model (PRT) (Pikovskii et al., 1978) has been elaborated in order to study interactions between “whistler waves” which propagate parallel to the magnetic field and lower hybrid waves in a plasma. Such interactions are among the important phenomena taking place in the ionosphere. This phenomenon can be modeled by means of a system of three coupled first-order ordinary differential equations in the variables x1 (t), x2 (t) and x3 (t), which give the the normal amplitude of the wave, the normal amplitude of the ion acoustic wave and the normal amplitude of the synchronous third wave produced, respectively. The amplitudes are assumed to be constant in space. The evolution equations of the (PRT) model may be written in dimensionless form:   V  

dx1 dt dx2 dt dx3 dt





f1 (x1 , x3 , x3 )





hx2 − ν1 x1 − x2 x3



      f2 (x1 , x3 , x3 )  =  hx1 − ν2 x2 + x1 x3  =      x1 x2 − ν3 x3 f3 (x1 , x3 , x3 )

(14.2)

where the amplitudes have been nondimensionalized; h is proportional to the amplitude of the electric field of the “whistler” and ν1 and ν2 are the damping decrements of the excited hybrid and acoustic waves normalized to the damping of the decay-induced third wave and where the functions fi are C ∞ functions in a compact E included in R3 and with values in R. According to Th. 8.1 and Corollary 8.1 it may be stated that the (PRT) model admits2 three fixed point : the origin O(0, 0, 0), I(xI , yI , zI ) and J(xJ , yJ , zJ ). Then, the Hessian of the flow curvature manifold evaluated at the equilibrium O is positive, i.e., Hφ(X ∗ ) > 0 while evaluated in I (resp. J) it is negative, i.e., Hφ(X ∗ ) < 0 the origin O is a saddle-node while fixed points I and J are saddle-foci (or center) according to Th. 8.4. Remark. By using the same theorems it can be shown that Chua’s model admits three foci as fixed points (Cf. Mathematica file). 2 Cf.

Mathematica file

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Chapter 15

Invariant Sets - Integrability

In the third part of this book many applications of the various concepts presented previously such as: invariants sets, center manifolds, normal forms, slow invariant manifolds, integrability and inverse problem are proposed in order to emphasize the flow curvature method.

15.1 MF 40


Let’s consider again the dynamical system (14.2): 

x˙1





f1 (x1 , x3 , x3 )





hx2 − ν1 x1 − x2 x3



           V   x˙2  =  f2 (x1 , x3 , x3 )  =  hx1 − ν2 x2 + x1 x3  f3 (x1 , x3 , x3 ) x˙3 x1 x2 − ν3 x3

(15.1)

It has been established (Giacomini et al., 1991) that this (PRT) model (15.1) admits invariant manifolds for certain values of the parameters. These results are presented in the following Table 15.1. Linear invariant manifolds First, let’s note that all invariant manifolds are decomposable quadrics except case 1-2. So, it may be checked (Cf. Mathematica file) according to Prop. 9.1 that each of these invariant manifolds is in factor of the flow curvature manifold associated with dynamical system (15.1). Let’s focus on case 4. With this set of parameters: (ν1 , ν2 = 0, ν3 = 0, h) the invariant manifold may be written: 2 (X) = y 2 − (z + h)2 = (y − z − h)(y + z + h) = ϕ2 (X)ϕ ϕ(X) 229

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So, let’s seek a multiplier of the form:

= (y − z − h)m (y + z + h)m = pm q m µ(X)

According to Eq. (6.3) we have: + T r (J) µ(X) = −((m − m )x + ν1 )µ(X) LV µ(X) is a multiplier provided that: m = m and ν1 = 0. Then, Thus, µ(X) taking ν1 = 0 the flow curvature manifold associated with dynamical system (15.1) with this new set of parameters, i.e. with (ν1 = 0, ν2 = 0, ν3 = 0, h) reads: = (y − z − h)(y + z + h)(x2 + (z − h)2 )Q(X) φ(X) is a polynomial of degree higher than one. So, it may be where Q(X) checked that a first integral of dynamical system (14.1) is: = (y − z − h)α1 (y + z + h)α2 (x2 + (z − h)2 )α3 ϕ(X) with α1 = α2 . Thus, Darboux theorem 9 and Th. 12.3 are found again since it has been stated that if the multiplier consists of two invariant planes the first integral is the product of these planes by a quadrics. Nonlinear invariant manifolds Now, let’s focus on case 1 since case 2 has been already studied in Ex. 9.6. If ν1 = ν2 = ν, and ν3 = 2ν the (PRT) model admits the nondecomposable quadrics: x2 + y 2 − 4hz as 2nd degree algebraic invariant manifold. According to Prop. 9.3 the 2nd extatic algebraic manifold reads: 

 x2 y2 z = L x2 L y 2 L (z) = −2νx2 y 2 (xy−2νz) x2 − y 2 − 4hz εd (X) V V V L2V x2 L2V y 2 L2V (z) = x2 − y 2 − 4hz its Lie derivative may be written: By posing ϕ(X) = −2ν(x2 − y 2 − 4hz) = k1 (X)ϕ( LV ϕ(X) X) = −2ν. So, x2 − y 2 − 4hz is a 2nd degree algebraic invariant where k1 (X) manifold for this dynamical system (Cf. Mathematica file).

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Table 15.1

231

Invariant manifolds of the (PRT)

model.

Invariant manifolds

(ν1 , ν2 , ν3 , h)

x2 + y 2 − 4hz x2 − y 2 + 2z 2 x2 + y 2 2 y − (z + h)2 x2 + (z − h)2 y2 − z 2 x2 + z 2

(ν, ν, 2ν, h) (ν, ν, ν, h) (ν, ν, ν3 , 0) (ν1 , 0, 0, h) (0, ν2 , 0, h) (ν1 , ν, ν, 0) (ν, ν2 , ν, 0)

1. 2. 3. 4. 5. 6. 7.

15.2 MF 41

Rikitake model

The Rikitake model (Rikitake, 1958) has been elaborated for describing the Earth’s magnetohydrodynamic dynamo and used by geophysicists as a conceptual mean for studying the time series of geomagnetic polarity reversals over geological time. This phenomenon can be modeled by means of a system of three coupled first-order ordinary differential equations in the real variables x1 (t), x2 (t) and x3 (t) which may be written in dimensionless form: 

x˙1





f1 (x1 , x3 , x3 )





−µx1 + x2 (x3 + β)



           V   x˙2  =  f2 (x1 , x3 , x3 )  =  −µx2 + x1 (x3 − β)  f3 (x1 , x3 , x3 ) α − x1 x2 x˙3

(15.2)

It has been established (Llibre and Zhang, 2000) that Rikitake model (15.2) admits invariant manifolds for certain values of the parameters. These results are presented in the following Table 15.2. Since all invariant manifolds are decomposable quadrics, it may be checked according to Prop. 9.1 that each of these invariant manifolds is in factor of the flow curvature manifold associated with dynamical system (15.2). Then, let’s focus on case 1. With this set of parameters: (0, 0, β) the flow curvature manifold associated with dynamical system (15.2) reads: = (y 2 + (z − β)2 )(x2 + (z + β)2 )Q(X) φ(X)

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232

is a polynomial of degree higher than one. It may be checked where Q(X) that each of these factors is first integral of dynamical system (15.2). Thus, both family of surfaces:

= (y 2 + (z − β)2 ) = C1 ϕ1 (X) = (x2 + (z + β)2 ) = C2 ϕ2 (X)

represent the general integral of system (15.2) with this set of parameters which is a family of space curves defined by their intersection.

Table 15.2

Invariant manifolds of the

Rikitake model.

1. 2. 3.

15.3 MF 42

Invariant manifolds

(µ, α, β)

x2 + z 2 ± 2βz x2 − y 2 x±y

(0, 0, β) (0, α, 0) (µ, α, 0)

Chua’s model

The cubic Chua’s circuit (Chua et al., 1986) is an electronic circuit comprising an inductance L1 , an active resistor R, two capacitors C1 and C2 , and a nonlinear resistor. Chua’s circuit can be accurately modeled by means of a system of three coupled first-order ordinary differential equations in the variables x (t), y (t) and z (t), which give the voltages in the capacitors C1 and C2 , and the intensity of the electrical current in the inductance L1 , respectively.    1   x˙ f1 (x, y, z) ε (z − k(x) − µx)  f2 (x, y, z)  =   V  y˙  = −βz z˙ f3 (x, y, z) −ax + y + bz

(15.3)

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The function k(x) describes the electrical response of the nonlinear resistor, i.e. its characteristics which is a “cubic” function is defined by: k(x) = c1 x3 + c2 x2 where the real parameters c1 and c2 are determined by the particular values of the circuit components. Parameter β = 1 for instance. It will be established that, for certain values of parameters, cubic Chua’s model (15.3) admits two invariant manifolds which may be directly deduced from its flow curvature manifold and then used to build a first integral . 1 + k2 flow curvature manifold of this system Setting a = 0 and b = k reads: = (kz + y) (z + ky) Q(X) =0 φ(X) is a polynomial of degree greater than one. Since we have: where Q(X) LV (kz + y) (z + ky) = b (kz + y) (z + ky) kz + y = 0 and z + ky = 0 are invariant manifolds according to Prop. 9.1. Thus, following Th. 12.1 if α1 and α2 are chosen such that α2 = −k 2 α1 , a first integral of Chua’s model reads: = (kz + y)α1 (z + ky)α2 = C ϕ(X)

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234

15.4 MF 43

DGeometry

Lorenz model

The purpose of the model established by Edward Lorenz (1963) was in the beginning to analyze the unpredictable behavior of weather. After having developed nonlinear partial derivative equations starting from the thermal equation and Navier-Stokes equations, Lorenz truncated them to retain only three modes. The most widespread form of the Lorenz model is as follows:       x˙ σ(y − x) f1 (x, y, z)  f2 (x, y, z)  =  −xz + rx − y  V  y˙  = f3 (x, y, z) z˙ xy − bz

(15.4)

where σ, b and r are real parameters. It has been established (Llibre and Zhang, 2002) that Lorenz model (15.4) admits invariant manifolds for certain values of the parameters. A part of these results is presented in the following Table 15.3. Let’s note that the first invariant manifold is a decomposable quadrics in the complex domain. So, it may be checked according to Prop. 9.1 that it is in factor of the flow curvature manifold associated with dynamical system (15.4) which may be written with parameters (1, 0, σ): = (y 2 + z 2 )Q(X) φ(X) is a polynomial of degree higher than one. Since we have: where Q(X) LV (y 2 + z 2 ) = −2(y 2 + z 2 ) y 2 + z 2 is an invariant manifold according to Prop. 9.1. Now, let’s focus on case 3. If b = σ = 1, Lorenz model (15.4) admits the non-decomposable quadrics: −rx2 +y 2 +z 2 as 2nd degree algebraic invariant manifold. According to Prop. 9.3 the 2nd extatic algebraic manifold reads: 

 y2 z2 x2   = LV x2 LV y 2 LV z 2  = 4x2 y 2 (xy − z)(rx2 − y 2 − z 2 ) εd (X)   2 2 2 2 2 2 LV x LV y LV z

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235

= −rx2 + y 2 + z 2 its Lie derivative may be written: By posing ϕ(X) = −2(−rx2 + y 2 + z 2 ) = k(X)ϕ( X) LV ϕ(X) = −2. So, −rx2 + y 2 + z 2 is a 2nd degree algebraic invariant where k(X) manifold for this dynamical system (Cf. Mathematica file).

Table 15.3

Invariant manifolds of the

Lorenz model.

1. 2. 3.

Invariant manifolds

(b, r, σ)

y2 + z 2 x2 − 2σz −rx2 + y 2 + z 2

(1, 0, σ) (2σ, r, σ) (1, r, 1)

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Chapter 16

Local Bifurcations

16.1 MF 44

Chua’s model

Let’s consider the Chua’s model (15.3) in which variables are such that: y → z and where it has been posed: a = b = −1, α = 1/ε, c1 = aαε, c2 = 0 and µ = cαε. Thus, Chua’s model becomes:       x˙ α y − ax3 − cx f1 (x, y, z)  f2 (x, y, z)  =   V  y˙  = x−y+z −βy f3 (x, y, z) z˙ By setting β = α, c = 0 this system in the eigenbasis reads:       x˙ y + a(x − z)3 (−1 + α)α3 f1 (x, y, z)  f2 (x, y, z)  =   y˙  =  V −a(x − z)3 α4 3 4 −z + a(x − z) α f3 (x, y, z) z˙ The center manifold z = h (x, y) solution to the partial Eq. (4.4) may be approximated arbitrarily closely as a Taylor series at (x, y) = (0, 0): z = h (x, y) = a20 x2 + a11 xy + a02 y 2 + a30 x3 + a21 x2 y + a12 xy 2 + a03 y 3 + · · · where it has been established (Albaga et al., 1999) that coefficients are: a20 = a11 = a02 = 0 , a30 = aα4 , a12 = −2a21 = −a03 = 6aα4 Flow curvature manifold1 φ (x, y, z) = 0 1 Cf.

associated

Mathematica file 237

with

this

system

reads:

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238

According to Eqs. (10.15):

1 ∂a10 = −6a20 leads to a20 = 0 (x,y)→(0,0) 2! ∂x 1 ∂a01 3 = lim = − a02 leads to a02 = 0 (x,y)→(0,0) 2! ∂y 2 ∂a10 ∂a01 = lim = lim = −6a11 leads to a11 = 0 (x,y)→(0,0) ∂y (x,y)→(0,0) ∂x

a20 = a02 a11

lim

Thus, the center manifold equation reads: z = h (x, y) = a30 x3 + a21 x2 y + a12 xy 2 + a03 y 3 + O x4 , y 4 So, this polynomial, truncated at order four, may be considered as homogeneous polynomial. Applying Euler’s theorem leads to: x

∂h ∂h +y = 3h (x, y) ∂x ∂y

Moreover, since this polynomial is supposed to be invariant with respect to the flow of the dynamical system, we have:

z˙ =

∂h ∂h x˙ + y˙ ∂x ∂y

⇔

 ∂h z˙ ∂h y˙   = −  ∂x x˙ ∂y x˙  ∂h z ˙ ∂h x˙   = − ∂y y˙ ∂x y˙

Plugging these equations in Euler ’s relations leads to:  ∂h   ˙ − xy) ˙ = 3h(x, y)  xz˙ + (xy ∂y  ∂h   y z˙ − (xy ˙ − xy) ˙ = 3h(x, y) ∂x Plugging h (x, y) in these equations and by solving order by order directly provides the unknown coefficients. In this case we find again: a30 = aα4 ; a21 = −3aα4 ; a12 = 6aα4 ; a03 = −6aα4

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Local Bifurcations

16.2 MF 45

DGeometry

239

Lorenz model

Let’s consider the classical model of Lorenz (1963) in its diagonal form:     x˙ f1 (x, y, z)  f2 (x, y, z)  V  y˙  = f3 (x, y, z) z˙      σε − σz 0 0 0 x x + σy   =  0 −1 − σ 0   y  + z−ε 1+σ (1 + σ) (x − y) 0 0 −b z where σ = 10, b = 8/3 and by setting r = 1 + ε in order to exhibit a pitchfork bifurcation in the vicinity of the origin when ε → 0. According to the Center Manifold Theorem (Ch. 4, Th. 4.1) the center manifold solution of the partial equation (4.4) can be approximated arbitrarily closely as a Taylor series at (x, y, z, ε) = (0, 0, 0, 0): y = h1 (x, ε) = a20 x2 + a11 xε + a02 ε2 + O x3 , ε3 z = h2 (x, ε) = b20 x2 + b11 xε + b02 ε2 + O x3 , ε3 These polynomials, truncated at order two, may be considered as homogeneous polynomials. Thus, Euler’s theorem may be applied to each of them: x

∂hi ∂hi +ε = 2hi (x, ε) with i = 1, 2 ∂x ∂ε

Moreover, these polynomials are supposed to be invariant with respect to the flow of the dynamical system, hence they may check respectively: y˙ −

∂h1 x˙ = 0 ∂x

⇔

∂h1 y˙ = ∂x x˙

z˙ −

∂h2 x˙ = 0 ∂x

⇔

∂h2 z˙ = ∂x x˙

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Plugging these equations in Euler ’s relations leads to: x

∂h1 y˙ +ε = 2h1 (x, ε) x˙ ∂ε

x

∂h2 z˙ +ε = 2h2 (x, ε) x˙ ∂ε

Plugging hi (x, ε) (i = 1, 2) in these equations and by solving order by order directly provides the unknown coefficients. In this case we find again: a20 = 0 ; a11 = −

1 (1 + σ)

2

; a02 = 0 ; b20 =

1 ; b11 = 0 ; b02 = 0 b

Finally, substituting these coefficients into Lorenz model in its diagonal form gives the vector field reduced to the center manifold: σx x˙ = 1+σ

1 2 ε − x + ··· b

and so, a pitchfork bifurcation occurs for ε → 0, i.e. r = 1. All center manifolds computed with these methods are completely identical with those obtained in the first part of this work (Ch. 4) and can also be used for detecting local bifurcations of codimension 1.

DGeometry

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DGeometry

Chapter 17

Slow-Fast Dynamical Systems – Singularly Perturbed Systems

This section is shared in three parts. The first can be considered as a tutorial in which piecewise linear models, singularly perturbed systems and slow fast dynamical systems of low-dimensions two and three are used in order to emphasize Flow Curvature Method. In the second part, Flow Curvature Method is applied to any high-dimensional dynamical systems. Then, in the third part, Flow Curvature Method is extended to non-autonomous dynamical systems. 17.1 17.1.1 MF 46

Piecewise Linear Models 2D & 3D Van der Pol piecewise linear model

In order to study the asymptotic behavior of relaxation oscillations V

1 x˙ (F (x) − y) f1 (x, y) ε = = y˙ f2 (x, y) x

(17.1)

where F (v) is the piecewise linear function:    −2 − v, F (v) = v,   2 − v,

v < −1 −1 ≤ v ≤ 1 v>1

with parameter ε = 0.01 and where the functions fi are infinitely differentiable with respect to all x, y and t, i.e. are C ∞ functions in a compact E included in R2 and with values in R.

241

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242

Thus, according to Prop. 11.1 the slow invariant manifold associated with Van der Pol piecewise linear model is given by the 1st flow curvature manifold : + + + ˙ ¨ + + +X ∧ X κ1 = + +3 + ˙ + +X +

⇔

+ + =+ ˙ ∧ X ¨ + φ(X) +X +=0

⇔

φ (x, y) = 0

˙ = J IM Since this Van der Pol model is piecewise linear we have: X where I is the fixed point and M any point of the phase plane. Moreover, ¨ = J X ˙ and so, the flow curvature manifold reads: according to Eq. (7.5) X + + =+ ∧ JX ˙ + φ(X) +J IM +=0

(17.2)

But, according to the second identity1 (A.15): Ja ∧Jb = Det (J) (a ∧b). Then, we have: + + =0 ˙ + ∧X = Det (J) + φ(X) + = Det (J) Π(X) +IM

(17.3)

is the osculating line passing through the points I(0, ±2). where Π(X) The Lie derivative of the flow curvature manifold reads: + + + + + ˙ + ¨ + ∧X = Det (J) + ∧ JX LV φ(X) + + = Det (J) +IM +IM

(17.4)

According to the third identity2 (A.16), i.e. Ja∧b+a∧Jb = T r (J) (a∧b) ∧ JX ˙ = T r (J) (IM ∧ X) ˙ and so, we have: IM + + = T r (J) Det (J) + ∧X ˙ + LV φ(X) +IM + = T r (J) φ(X)

(17.5)

So, according to Darboux invariance theorem and Prop. 11.2 the slow manifold of Van der Pol piecewise linear model given by the flow curvature manifold , i.e. each osculating line is globally invariant (Cf. Def 9.1). With parameter ε = 0.01 both osculating lines equation read:

y±2+x 1 Cf. 2 Cf.

Appendix Appendix

1+

√ 1 − 4ε =0 2

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243

Y 2

1

0

X

1

2 3

Fig. 17.1

17.1.2 MF 47

2

1

0

1

2

3

Van der Pol piecewise linear model slow invariant manifold

Chua’s piecewise linear model

Let’s consider the Chua’s model       x˙ f1 (x, y, z) α (y − x − k (x))       f2 (x, y, z) =  y˙  =  V x−y+z       z˙ f3 (x, y, z) −βy

(17.6)

The function k (x) describes the electrical response of the nonlinear resistor, i.e. its characteristics which is a now piecewise linear function defined by:    bx + a − b, k(x) = ax,   bx − a + b,

x≥1 |x| ≤ 1 x ≤ −1

where the real parameters α and β determined by the particular values of the circuit components are in a standard model α = 1/9, β = 100/7, a = −8/7 and b = −5/7 and where the functions fi are C ∞ functions in a compact E included in R3 and with values in R.

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244

Thus, according to Prop. 11.1 slow invariant manifold associated with Chua’s piecewise linear model is given by 2nd flow curvature manifold: ... ... ˙ · (X ¨ ∧ X) X = 0 ⇔ φ (x, y, z) = 0 =X ˙ · (X ¨ ∧ X) κ2 = + +2 ⇔ φ(X) + ˙ ¨+ +X ∧ X + ˙ = J IM where I Since this Chua model is piecewise linear we have: X is the fixed point and M any point of the phase space. Moreover, according ¨ = J X ˙ and since the time derivative of the functional to (7.5) we have: X dJ jacobian matrix is zero: = 0, over-acceleration (or jerk) reads: dt ... = JX ¨ + dJ X ˙ = J X. ¨ So, the flow curvature manifold reads: X dt ¨ = 0 ˙ ∧ J X) = J IM · (J X φ(X)

(17.7)

Then, identity3 (A.15) Ja.(Jb ∧ Jc) = Det (J) a.(b ∧ c) leads to: = Det (J) IM · (X ˙ ∧ X) ¨ = 0 φ(X)

(17.8)

= IM ·(X ˙ ∧ X) ¨ = 0 is the osculating plane passing through where, Π1,2 (X) the fixed point I1 (resp. I2 ). ... = JX ¨ reads, taking into account that X The Lie derivative of φ(X) ... = Det (J) IM · (X ˙ ∧ J X) ¨ = 0 = Det (J) IM · (X ˙ ∧ X) LV φ(X)

(17.9)

Identity (A.16) Ja.(b ∧ c) + a.(Jb ∧ c) + a.(b ∧ Jc) = T r (J)a.(b ∧ c) · (X ˙ ∧ J X) ¨ = T r (J) IM · (X ˙ ∧ X) ¨ and so leads to: IM = T r [J] φ(X) LV φ(X) Thus, according to Darboux invariance theorem and Prop. 11.2 slow manifold associated with Chua’s piecewise linear model given by flow curvature manifold , i.e. each osculating plane is globally invariant (Cf. Def 9.1). 3 Cf.

Appendix

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DGeometry


Fig. 17.2

245

Chua’s piecewise linear model slow invariant manifold

With this set of parameters: α = 1/9, β = 100/7, a = −8/7, b = −5/7, Π1,2 hyperplanes equations passing through fixed point I1,2 (∓3/2, 0, ±3/2) read: = 2.8759x − 3.9421y + z ± 2.8139 = 0 Π1,2 (X) 17.2

Singularly Perturbed Systems 2D & 3D

17.2.1 MF 48

FitzHugh-Nagumo model

The FitzHugh-Nagumo model (FitzHugh, 1961; Nagumo et al., 1962) V

1 x˙ f1 (x, y) = ε (x(1 − x)(x + a) − y + b) = y˙ f2 (x, y) x − cy

(17.10)

x denotes the membrane potential, y is a recovery variable and b is the magnitude of stimulus current with parameter ε = 0.01, a = 0.1, b = 0.01 and c = 0.5 and where the functions fi are infinitely differentiable with respect to all x, y and t, i.e. are C ∞ functions in a compact E included in R2 and with values in R.

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246

Y 0.4

0.3

0.2

0.1

0.0

X

0.1

0.2

0.5

Fig. 17.3

0.0

0.5

1.0

FitzHugh-Nagumo model slow invariant manifold

Thus, according to Prop. 11.1 the slow invariant manifold (Fig. 17.3, blue curve) associated with FitzHugh-Nagumo model is given by the 1st flow curvature manifold (Cf. Mathematica file). The Lie derivative of the flow curvature manifold may be written as: = T r(J)φ(X) + P (X) ψ(X) LV φ(X)

2

= x(1 − x)(x + a) + b − y = 0 is the singular approximation where ψ(X) a polynomial of degree greater than one. (Fig. 17.3, green curve) and P (X) So, in the vicinity of the singular approximation we have: = T r(J)φ(X) LV φ(X)

∈φ ∀X

and

∈ψ X

Thus, according to Darboux invariance theorem and Prop. 11.2 the slow manifold of FitzHugh-Nagumo model given by the flow curvature manifold is locally invariant (Cf. Def 9.2).

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DGeometry


17.2.2 MF 49

247

Chua’s model

Let’s consider again cubic Chua’s model (15.3) :     1  3 2 x˙ f1 (x, y, z) ε z − c1 x − c2 x − µx f2 (x, y, z) =   V y˙  = −βz f3 (x, y, z) z˙ −ax + y + bz with the following set of parameters: c1 = 44/3 ; c2 = 41/2 ; µ = 6.94 ; β = 1 ; ε = 0.01 ; a = 0.7 and b = 0.24. Chua’s model (15.3) is a singularly perturbed system the slow manifold of which may be provided either by the Geometric Singular Perturbation Method or by Flow Curvature Method (Cf. Ch. 11). Thus, according to Prop. 11.1 the slow invariant manifold associated with Chua’s model is given by the 2nd flow curvature manifold (Cf. Mathematica file).

Fig. 17.4

Chua’s cubic model slow invariant manifold

Moreover, it may be stated, in the vicinity of the singular approximation = z − c1 x3 − c2 x2 − µx = 0, that the functional jacobian defined by ψ(X) matrix is locally stationary and so, = T r(J)φ(X) LV φ(X)

∈φ ∀X

and

∈ψ X

Thus, according to Darboux invariance theorem and Prop. 11.2 the slow manifold of cubic Chua’s model given by the flow curvature manifold is locally invariant (Cf. Def 9.2).

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248

17.3

Slow Fast Dynamical Systems 2D & 3D

17.3.1 MF 50

DGeometry

Brusselator model

Studying an hypothetical V

x˙ f1 (x, y) 1 − x − βx + αx2 (y − x) = = y˙ f2 (x, y) 1−x

(17.11)

x and y denotes concentrations with parameter α = 5, β = 7 and where the functions fi are infinitely differentiable with respect to all x, y and t, i.e. are C ∞ functions in a compact E included in R2 and with values in R. According to Grasman (1987) for β > α + 1 " 1 with β − α = O (1) the chemical oscillation turns into relaxation oscillation. Thus, the Brusselator may be considered as slow-fast dynamical systems although its has no small multiplicative parameter in its velocity vector field and so no singular approximation. Thus, according to Prop. 11.1 the slow invariant manifold associated with Brusselator’s model is given by the 1st flow curvature manifold (Cf. Mathematica file). 5.0

4.5

4.0

3.5

3.0

2.5

2.0

Fig. 17.5

0

1

2

3

4

5

Brusselator’s model slow invariant manifold

Moreover, it may be stated that in the vicinity of the flow curvature manifold both flow curvature manifold (Cf. Fig. 16.5 in blue) and its Lie derivative (Cf. Fig. 16.5 in pink) are merged. Thus, according to Darboux invariance theorem and Prop. 11.2 the slow manifold of Brusselator’s model is locally invariant (Cf. Def 9.2).

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17.3.2 MF 51

DGeometry

249


Let’s consider again the Pikovskii-Rabinovich-Trakhtengerts model (       f1 (x, y, z) x˙ hy − ν1 x + yz       f2 (x, y, z) = hx − ν2 y − xz  y˙  = V       z˙ f3 (x, y, z) xy − ν3 z It has been recently established (Llibre et al., 2008) that this dynamical system exhibits a four wings butterfly chaotic attractor for the following set of parameters: h = 0.04 ; ν1 = 1.5 ; ν2 = −0.3 and ν3 = 1.67. Although the (PRT) model is not singularly perturbed it can be considered as a slow fast dynamical system since according to Def. 5.2 it may be shown (numerically) that its functional Jacobian matrix exhibits a fast eigenvalue. Thus, according to Prop. 11.1 the slow invariant manifold associated with the (PRT) model is given by the flow curvature manifold (Cf. Mathematica file).

Fig. 17.6

(PRT) model slow invariant manifold

Moreover, it may be stated that in the vicinity of the flow curvature manifold both flow curvature manifold and its Lie derivative are merged. Thus, according to Darboux invariance theorem and Prop. 11.2 the slow manifold of (PRT) model is locally invariant (Cf. Def 9.2).

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17.3.3 MF 52



Rikitake model

Let’s consider again the Rikitake model

      f1 (x, y, z) x˙ −µx + yz       f2 (x, y, z) = −µy + x(z − a) y˙  = V       z˙ α − xy f3 (x, y, z)

with the following set of parameters: µ = 2 ; a = 5 and α = 1. Although the Rikitake model is not singularly perturbed it can be considered as a slow fast dynamical system since according to Def. 5.2 it may be shown (numerically) that its functional Jacobian matrix exhibits a fast eigenvalue. Thus, according to Prop. 11.1 the slow invariant manifold associated with the Rikitake model is given by the flow curvature manifold (Cf. Mathematica file).

Fig. 17.7

Rikitake model slow invariant manifold

Moreover, it may be stated that in the vicinity of the flow curvature manifold both flow curvature manifold and its Lie derivative are merged. Thus, according to Darboux invariance theorem and Prop. 11.2 the slow manifold of Rikitake model is locally invariant (Cf. Def 9.2).

DGeometry

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17.4

251

Piecewise Linear Models 4D & 5D

17.4.1 MF 53

DGeometry

Chua’s fourth-order piecewise linear model

The piecewise linear fourth-order Chua resistors R and R1 , two capacitors C1 and C2 , and a nonlinear resistor. Fourth-order Chua’s circuit can be accurately modeled by means of a system of four coupled first-order ordinary differential equations in the variables x1 (t), x2 (t), x3 (t) and x4 (t), which give the voltages in the capacitors C1 and C2 , and the intensities of the electrical current in the inductance L1 and L2 , respectively. These equations called global unfolding of Chua’s circuit are written in a dimensionless form:    α1 (x3 − k (x1 )) f1 (x1 , x2 , x3 , x4 )       f2 (x1 , x2 , x3 , x4 )  α2 x2 − x3 − x4  x˙2        V   =  =  f3 (x1 , x2 , x3 , x4 ) β1 (x2 − x1 − x3 ) x˙3  

x˙1





f4 (x1 , x2 , x3 , x4 )

x˙4

(17.12)

β2 x2

Function k (x1 ) describes the electrical response of the nonlinear resistor, i.e. its characteristics which is a piecewise linear function defined by:    bx1 + a − b, k (x1 ) = ax1 ,   bx1 − a + b,

x1 ≥ 1 |x1 | ≤ 1 x1 ≤ −1

where the real parameters αi and βi determined by the particular values of the circuit components are in a standard model α1 = 2.1429, α2 = −0.18, β1 = 0.0774, β2 = 0.003 a = −0.42, b = 1.2 and where the functions fi are infinitely differentiable with respect to all xi , and t, i.e. are C ∞ functions in a compact E included in R4 and with values in R. Thus, according to Prop. 11.1 the slow invariant manifold 4 associated with Chua’s piecewise linear model is given by the 3rd flow curvature manifold : =V · (γ ∧ γ˙ ∧ γ¨ ) = 0 φ(X) (n+2)

(n) . =X ˙ and γ = X in which it has been posed: V 4 Cf.

Mathematica file

(17.13)

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252

(n)

= J (n)γ . Piecewise linear feature enables to state that: γ = J (n+1) V = J IM , the slow manifold So, according to the fact that as previously: V equation (17.13) reads: = J IM · (J V ∧ Jγ ∧ J γ˙ ) = 0 φ(X)

(17.14)

= Det (J) a.(b ∧ c ∧ d) leads to: Identity5 (A.15) Ja.(Jb ∧ Jc ∧ J d) = Det (J) IM · V ∧ γ ∧ γ˙ = 0 φ(X)

(17.15)

= IM · (V ∧ γ ∧ γ˙ ) = 0 is the osculating hyperplane passing where Π1,2 (X) through the fixed point I1 (resp. I2 ). Taking into account that γ˙ = Jγ and γ¨ = J γ˙ the Lie derivative of φ(X) reads: = Det (J) IM · (V ∧ γ ∧ γ¨) LV φ(X) · (V ∧ γ ∧ J γ˙ ) = Det (J) IM

(17.16)

The identity (A.16) established in appendix leads to: · (V ∧ γ ∧ J γ˙ ) = T r (J) IM · (V ∧ γ ∧ γ˙ ) IM and = T r [J] φ(X) LV φ(X) So, according to Darboux invariance theorem and Prop. 11.2 the slow manifold associated with Chua’s fourth-order piecewise linear model given by the flow curvature manifold, i.e. each osculating hyperplane is globally invariant (Cf. Def 9.1). With this set of parameters: α1 = 2.1429, α2 = −0.18, β1 = 0.0774, β2 = 0.003, a = −0.42 and b = 1.2, Π1,2 hyperplanes equations passing through fixed points I1,2 (∓0.73, 0, ±0.73, ∓0.73) read: = 1.8861x1 + 0.04744x2 − 1.6461x3 + 0.01895x4 ± 2.6149 = 0 Π1,2 (X) 5 Cf.

Appendix

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DGeometry


Fig. 17.8

17.4.2 MF 54

253

Chua’s model invariant hyperplanes in (x1 , x2 , x3 ) phase space

Chua’s fifth-order piecewise linear model

The piecewise linear fifth-order Chua electronic circuit consists of two inductances L1 and L2 , two linear resistors R and R1 , three capacitors C1 , C2 and C3 , and a nonlinear resistor. Fifth-order Chua’s circuit can be accurately modeled by means of a system of five coupled first-order ordinary differential equations in the variables x1 (t), x2 (t), x3 (t), x4 (t) and x5 (t), which give the voltages in the capacitors C1 , C2 and C3 , and the intensities of the electrical current in the inductance L1 and L2 , respectively. These equations called global unfolding of Chua’s circuit are written in a dimensionless form: 

x˙1





    x˙2        x˙3  = V      x˙    4 x˙5

α1 (x2 − x1 − k (x1 )) α2 x1 − x2 + x3 β1 (x4 − x2 ) β2 (x3 + x5 ) γ2 (x4 + γ1 x5 )

        

(17.17)

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DGeometry


Function k (x1 ) describes the electrical response of the nonlinear resistor, i.e. its characteristics which is a piecewise linear function defined by:    bx1 + a − b, k (x1 ) = ax1 ,   bx1 − a + b,

x1 ≥ 1 |x1 | ≤ 1 x1 ≤ −1

where real parameters αi , βi and γi are determined by the particular values of the circuit components and where the functions fi are infinitely differentiable with respect to all xi , and t, i.e. are C ∞ functions in a compact E included in R5 and with values in R. Thus, according to Prop. 11.1 slow invariant manifold 6 associated with Chua’s piecewise linear model is given by 4th flow curvature manifold : ... =V · (γ ∧ γ˙ ∧ γ¨ ∧ γ ) = 0 φ(X)

(17.18)

So, according to Darboux invariance theorem and Prop. 11.2 the slow manifold associated with Chua’s fifth-order piecewise linear model given by the flow curvature manifold, i.e. each osculating hyperplane is globally invariant (Cf. Def 9.1). With the following parameters set: α1 = 9.934, α2 = 1, β1 = 14.47, β2 = −406.5, γ1 = −0.0152, γ2 = 41000, a = −1.246 and b = −0.6724, Π1,2 hyperplanes equations passing through fixed point I1,2 (∓1.83, ∓0.027, ±1.8, ∓0.027, ∓1.8) read: = −2.63x1 + 3.78x2 − 0.846x3 − 0.000454x4 + 0.000298x5 ∓ 3.205 Π1,2 X

6 Cf.

Mathematica file

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Fig. 17.9

17.5

Chua’s model invariant hyperplanes in (x1 , x2 , x3 ) phase space

Singularly Perturbed Systems 4D & 5D

17.5.1 MF 55

255

Chua’s fourth-order cubic model

The fourth-order cubic Chua but while replacing the piecewise linear function by a smooth cubic nonlinear.    α1 (x3 − k (x1 )) f1 (x1 , x2 , x3 , x4 )       f2 (x1 , x2 , x3 , x4 )  α2 x2 − x3 − x4  x˙2        V   =  =  f3 (x1 , x2 , x3 , x4 ) β1 (x2 − x1 − x3 ) x˙3  

x˙1

x˙4





f4 (x1 , x2 , x3 , x4 )

(17.19)

β2 x2

The function kˆ (x1 ) describing the electrical response of the nonlinear resistor is an odd-symmetric function similar to the piecewise linear nonlinearity k (x1 ) for which the parameters c1 = 0.3937 and c2 = −0.7235 are determined while using least-square method (Tsuneda, 2005) and which characteristics is defined by: kˆ (x1 ) = c1 x31 + c2 x1

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256

The real parameters αi and βi determined by the particular values of the circuit components are in a standard model α1 = 2.1429, α2 = −0.18, β1 = 0.0774, β2 = 0.003, c1 = 0.3937 and c2 = −0.7235 and where the functions fi are infinitely differentiable with respect to all xi , and t, i.e. are C ∞ functions in a compact E included in R4 and with values in R. Thus, according to Prop. 11.1 the slow invariant manifold 7 associated with Chua’s fourth-order cubic model is given by the 3rd flow curvature manifold : ... =V · (γ ∧ γ˙ ∧ γ¨ ∧ γ ) = 0 φ(X)

(17.20)

Then, it may be stated that in the vicinity of the flow curvature manifold both flow curvature manifold and its Lie derivative are merged. Thus, according to Darboux invariance theorem and Prop. 11.2 the slow manifold of Chua’s fourth-order cubic model is locally invariant (Cf. Def 9.2).

Fig. 17.10

7 Cf.

Chua’s model slow invariant manifold in (x1 , x2 , x3 ) phase space

Mathematica file

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17.5.2 MF 56

257

Chua’s fifth-order cubic model

The fifth-order cubic Chua the piecewise linear function by a smooth cubic nonlinear. 

x˙1





   x˙2         x˙3  = V     x˙    4  x˙5

α1 (x2 − x1 − k (x1 )) α2 x1 − x2 + x3 β1 (x4 − x2 ) β2 (x3 + x5 )

        

(17.21)

γ2 (x4 + γ1 x5 )

The function kˆ (x1 ) describing the electrical response of the nonlinear resistor is an odd-symmetric function similar to the piecewise linear nonlinearity k (x1 ) for which the parameters c1 = 0.1068 and c2 = −0.3056 are determined while using least-square method (Tsuneda, 2005) and which characteristics is defined by: kˆ (x1 ) = c1 x31 + c2 x1 The real parameters αi , βi and γi determined by the particular values of the circuit components are: α1 = 9.934, α2 = 1, β1 = 14.47, β2 = −406.5, γ1 = −0.0152, γ2 = 41000, a = −1.246, b = −0.6724, c1 = 0.1068 and c2 = −0.3056 and where the functions fi are infinitely differentiable with respect to all xi , and t, i.e. are C ∞ functions in a compact E included in R5 and with values in R. Thus, according to Prop. 11.1 slow invariant manifold 8 associated with Chua’s fifth-order cubic model is given by 4th flow curvature manifold : ... .... =V · (γ ∧ γ˙ ∧ γ¨ ∧ γ ∧ γ ) = 0 φ(X)

(17.22)

Then, it may be stated that in the vicinity of the flow curvature manifold both flow curvature manifold and its Lie derivative are merged. Thus, according to Darboux invariance theorem and Prop. 11.2 the slow manifold of Chua’s fifth-order cubic model is locally invariant (Cf. Def 9.2). 8 Cf.

Mathematica file

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258

Fig. 17.11

17.6

Chua’s slow invariant manifold in (x1 , x2 , x3 ) phase space

Slow Fast Dynamical Systems 4D & 5D

17.6.1 MF 57

DGeometry

Homopolar dynamo model

As previously pointed out a simple model     x˙1 ρ(x2 − x1 )     x˙  (µ + ακ )x1 − (1 + µ)x2 − x1 x3 − βx4    2 =  V     x ˙ αx ((1 + µ)x − µx ) − κx  3   1 2 1 3 x˙4

(17.23)

−λx4 + (1 + µ)x2 − µx1

The real parameters are: α = 60, β = 9.6, λ = 1.2, µ = 0.5, ρ = 16 and κ = 1 and where the functions fi are infinitely differentiable with respect to all xi , and t, i.e. are C ∞ functions in a compact E included in R4 and with values in R. Although the homopolar dynamo model is not singularly perturbed it can be considered as a slow fast dynamical system since according to Def. 5.2 it may be shown (numerically) that its functional Jacobian matrix exhibits a fast eigenvalue. Thus, according to Prop. 11.1 the slow invariant manifold associated with the homopolar dynamo model is given by the 3rd

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259

flow curvature manifold (Cf. Mathematica file): ... =V · (γ ∧ γ˙ ∧ γ¨ ∧ γ ) = 0 φ(X)

Fig. 17.12

(17.24)

Dynamo model slow invariant manifold in (x1 , x2 , x3 ) phase space

Then, it may be stated that in the vicinity of the flow curvature manifold both flow curvature manifold and its Lie derivative are merged. Thus, according to Darboux invariance theorem and Prop. 11.2 the slow manifold of homopolar dynamo model is locally invariant (Cf. Def 9.2).

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17.6.2 MF 58


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Mofatt model

In the beginning of the nineties H. K. Mofatt (1993) By rescaling the variables suitably and shifting the origin, we obtain the model equations: 

x˙1





α (−ηx1 + ωx2 x3 )



     −ηx2 + ωx2 x3  x˙2          V κ (x4 − x3 − x1 x3 ) x˙3  =      −x + ξx − x x  x˙  3 3 5  4  4 x˙5

(17.25)

−x5 + x3 x4

where ξ, κ, η are measures of thermal forcing, frictional resistance and magnetic diffusion respectively, and α, ω are geometrical parameters. Let’s recall also the meaning of the dependent variables: x1 is the flux difference, x2 is the total flux, x3 is the angular velocity, and x4 , x5 give the temperature in the fluid loop. Although the Mofatt’s model is not singularly perturbed it can be considered as a slow fast dynamical system since according to Def. 5.2 it may be shown (numerically) that its functional Jacobian matrix exhibits a fast eigenvalue. Thus, according to Prop. 11.1 the slow invariant manifold associated with the Mofatt’s model is given by the 4th flow curvature manifold (Cf. Mathematica file). ... .... =V · (γ ∧ γ˙ ∧ γ¨ ∧ γ ∧ γ ) = 0 φ(X)

(17.26)

Then, it may be stated that in the vicinity of the flow curvature manifold both flow curvature manifold and its Lie derivative are merged. Thus, according to Darboux invariance theorem and Prop. 11.2 the slow manifold of Mofatt’s model is locally invariant (Cf. Def 9.2).

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Fig. 17.13

17.6.3 MF 59

261

Mofatt model slow invariant manifold in (x1 , x2 , x3 ) phase space

Magnetoconvection model

A fifth-order system for magnetoconvection describe nonlinear coupling between Rayleigh-Bernard convection and an external magnetic field. This type of system was first presented by Veronis (1966) in studying a rotating fluid. The fifth-order system of magnetoconvection is a straightforward extension of the Lorenz model for the Boussinesq convection interacting with the magnetic field. The fifth-order autonomous system of magnetoconvection is given as follows:

    σ −x1 + rx2 − qx4 1 + ςω(3−ω) x˙1 2 (4−ω) x5       x˙2  −x2 + x1 − x1 x3           V x˙3  =  ω (−x + x x ) 3 1 2      x˙  ω   −ς (x − x ) − x x  4 4 1 1 5   ς(4−ω) x˙5 −ς (4 − ω) (x5 − x1 x4 )

(17.27)

where x1 (t) represents the first-order velocity perturbation, while x2 (t), x3 (t), x4 (t) and x5 (t) are measures of the first- and the second-order perturbations to the temperature and to the magnetic flux function, respectively. With the five real parameters where ς = 0.09683 is the magnetic Prandtl number (the ratio of the magnetic to the thermal diffusivity), σ = 1 is the Prandtl number, r = 14.47 is a normalized Rayleigh number, q = 5 is a normalized Chandrasekhar number, and ω = 0.1081 is a geometrical parameter and where the functions fi are infinitely differentiable with re-

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spect to all xi , and t, i.e. are C ∞ functions in a compact E included in R5 and with values in R. Thus, according to Prop. 11.1 the slow invariant manifold associated with fifth-order magnetoconvection model is given by the 4th flow curvature manifold : ... .... =V · (γ ∧ γ˙ ∧ γ¨ ∧ γ ∧ γ ) = 0 φ(X)

(17.28)

Although the magnetoconvection model is not singularly perturbed it can be considered as a slow fast dynamical system since according to Def. 5.2 it may be shown (numerically) that its functional Jacobian matrix exhibits a fast eigenvalue. Thus, according to Prop. 11.1 the slow invariant manifold associated with the magnetoconvection model is given by the flow curvature manifold (Cf. Mathematica file).

Fig. 17.14

Magnetoconvection slow invariant manifold in (x1 , x2 , x3 ) phase space

Then, it may be stated that in the vicinity of the flow curvature manifold both flow curvature manifold and its Lie derivative are merged. Thus, according to Darboux invariance theorem and Prop. 11.2 the slow manifold of magnetoconvection model is locally invariant (Cf. Def 9.2).

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17.7

263

Slow manifold gallery

In this section two examples of slow invariant manifolds of chaotic attractors are presented. The first is a chemical kinetics model (Gaspard and Nicolis, 1983). The second is a neuronal bursting model (Hindmarsh and Rose, 1984).

(a) Fig. 17.15

(b)

(a) Chemical kinetics model. (b) Neuronal bursting model.

A gallery of slow invariant manifolds is accessible at: http://ginoux. univ-tln.fr 17.8 MF 60

Forced Van der Pol model

In this last section it will be shown that Flow Curvature Method be written as:       3 εx˙ f (x, y, z) x + y − x3   g (x, y, z)  =   y˙  = V −x + aSin (2πθ) h (x, y, z) θ˙ ω

(17.29)

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264

A variable changes may transform this non-autonomous system into a slow-fast autonomous one which reads: 

x˙1





f1 (x1 , x2 , x3 , x4 )





      f2 (x1 , x2 , x3 , x4 )    x˙2   =  = V      f x ˙ (x , x , x , x )  3 1 2 3 4    3  x˙4

f4 (x1 , x2 , x3 , x4 )

1 ε

x1 + x2 −

x32 3

−x1 + ax3 Ωx4

      

(17.30)

−Ωx3

where ε = 0.002, a = 1.8, ω = 1.342043 and Ω = 2πω. Although this transformation increases the dimension of the system (17.29) the Flow Curvature Method enables, according to Prop. 11.1, to directly compute the slow manifold analytical equation associated with system (17.30). the equation of which reads: ... =V · (γ ∧ γ˙ ∧ γ¨ ∧ γ ) = 0 φ(X)

Fig. 17.16

(17.31)

Forced Van der Pol model slow invariant manifold in (x1 , x2 , x3 ) phase space

Then, it may be stated that in the vicinity of the flow curvature manifold both flow curvature manifold and its Lie derivative are merged. Thus, according to Darboux invariance theorem and Prop. 11.2 the slow manifold of forced Van der Pol model is locally invariant (Cf. Def 9.2).

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Discussion

In this work a new approach which consists in applying Differential Geometry to Dynamical Systems and called Flow Curvature Method has been presented. By considering the trajectory curve, integral of any n-dimensional dynamical system, as a curve in Euclidean n-space, the curvature of the trajectory curve, i.e. curvature of the flow has been analytically computed enabling thus to define a manifold called: flow curvature manifold. It has been stated that, since such manifold only involves the time derivatives of the velocity vector field and so, contains information about the dynamics of the system, it enables to find again the main features of the dynamical system studied. Thus, fixed points stability, invariant sets, centre manifold , normal forms, local bifurcations, slow invariant manifold and integrability of any n-dimensional dynamical systems have been deduced from the flow curvature manifold, i.e. according to the Flow Curvature Method . The concepts of global invariance and local invariance has been (re)defined from Darboux invariance theorem. So, it has been stated that flow curvature manifold also enabled to “detect” linear invariant manifolds of any n-dimensional dynamical systems which may be used to build first integrals of these systems. For nonlinear invariant manifolds identity between flow curvature manifold and the so-called extatic manifolds has also been stated. It has been established that the Flow Curvature Method enabled to easily compute the coefficients of the centre manifold approximation of any n-dimensional dynamical systems according to global invariance of the flow curvature manifold. Then, a link between normal forms of dynamical systems and “normal forms” of flow curvature manifold has been highlighted. Such a link enabled to directly compute the normal form of a dynamical system starting from its flow curvature manifold .

265

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For singularly perturbed systems comprising a small multiplicative parameter ε in factor in their velocity vector field, identity between Geometric Singular Perturbation Theory and Flow Curvature Method has been pointed out up to suitable order in ε. Moreover, identity between Fenichel’s invariance and Darboux invariance theorem has been also demonstrated. Thus, it has been stated that the 1st flow curvature manifold associated with a two-dimensional dynamical system directly provided the same first order approximation in ε of the slow invariant manifold as those given by Geometric Singular Perturbation Theory while the 2nd flow curvature manifold associated with a three-dimensional dynamical system directly provided the same second order approximation in ε of the slow invariant manifold. High orders approximation of the slow invariant manifold may be simply obtained by replacing the flow curvature manifold by its successive Lie derivatives. The main difference between Flow Curvature Method and the so-called Geometric Singular Perturbation Theory being that flow curvature manifold directly provides the slow invariant manifold analytical equation of any n-dimensional slow-fast dynamical systems not only singularly perturbed but also for non-singularly perturbed as exemplified with Lorenz model. Invariance of the flow curvature manifold , i.e. of the slow manifold has then been stated according to Darboux invariance theorem. It has been established that Darboux theory of integrability conjugated with Flow Curvature Method enabled to build first integrals of dynamical systems. By considering that the only knowledge about a polynomial dynamical system is its flow curvature manifold, it has been stated that one may find a family of vector field comprising this polynomial dynamical system solving thus the inverse problem. Then, it has also been shown that Flow Curvature Method may be easily applied to any n-dimensional autonomous dynamical systems singularly perturbed or non-singularly perturbed, i.e. slow-fast autonomous dynamical systems such as Lorenz, Rikitake, (PRT) models, . . . , or any n-dimensional non-autonomous dynamical systems singularly perturbed or non-singularly perturbed such as forced Van der Pol model.

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Discussion

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267

In this work it has been established that a new geometric approach called Flow Curvature Method is consistent with the analytical Dynamical Systems Theory and so, it seems that flow curvature manifold plays a very important role in the study of dynamical systems as well as centre manifold . Many perspectives can be given to this work. At first, the inverse problem related to symmetry should be deeply studied. Then, a classification of dynamical systems by using local metrics properties such as curvatures could bring many information about the structure of its attractor. Thus, it may be stated1 that the non stationary part of the flow curvature manifold represents the envelope of the location of the points where the curvature of the trajectory curve vanishes and so enables to exhibit the attractor structure. As an example, the non stationary part of the 2nd flow curvature manifold associated with the Chua’s cubic model Eq. (15.3) has been plotted in the figure below which consists of two ellipsoids, two elliptic paraboloids and one plane. Starting from the fixed point I+ in the center of the figure one can see that this point is connected with each others with an ellipsoid which indicates that this is a repulsive saddle-focus. Then, reaching the others fixed points the origin O at the top and I− at the bottom a paraboloid appears which means that this is an attractive saddle-focus.

Chua’s cubic model attractor structure in (x1 , x2 , x3 ) phase space 1 Cf.

publication with Professor C. Letellier. In press.

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Thus, it may be considered, according to the quotation introducing the approach developed in this book that: “. . . every time the system absorbs energy the curvature of its trajectory decreases and viceversa. . . ” This could lead to the idea that the trajectory curve is moving according to the least-action principle. At last, but not at least the problem of the metrics in which the space is embedded is the most fundamental problem for improving the Flow Curvature Method.

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Appendix A

In this appendix main features inherent to Dynamical Systems Theory such as Jordan form, fractal dimension, homeomorphism, diffeomorphism and to Differential Geometry like Lie derivative, Gram-Schmidt process or Frénet frame are recalled. The Generalized Tangent Linear System Approximation is also presented and formalized.

A.1

Lie derivative

(t) the Let φ a C 1 function defined in a compact E included in R and X integral of the dynamical system defined by (2.1). The Lie derivative is defined as follows:

· ∇φ = LV φ = V

n ∂φ dφ x˙ i = ∂xi dt i=1

Notation. dn φ LnV φ = LV Ln−1 φ =

V dtn Example A.1. d2 φ L2V φ = LV LV φ = 2 dt

;

L3V φ = LV LV LV ϕ X

=

d3 φ dt3

269

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A.2


Hessian

In mathematics, the Hessian matrix is the square matrix of secondorder partial derivatives of a function. Given the real-valued function f (x1 , x2 , . . . , xn ) if all second partial derivatives of f exist, then the Hessian matrix of f is the matrix: 2 ∂ f2 ∂x1 ∂2f H (f ) = ∂x2 ∂x1 .. . ∂2f

∂2 f ∂x1 ∂x2 ∂2 f ∂x22

.. .

∂2 f ∂xn ∂x1 ∂xn ∂x2

···

∂2 f ∂x1 ∂xn

··· .. .

∂2 f ∂x2 ∂xn

...

.. .

∂2 f ∂x2n

(Some mathematicians define the Hessian as the determinant of the above matrix). The term ”Hessian” was coined by James Joseph Sylvester, named for German mathematician Ludwig Otto Hesse, who had used the term ”functional determinants”. A.3

Jordan form

In linear algebra, Jordan normal form (often called Jordan canonical form) shows that a given square matrix M over a field K containing the eigenvalues of M can be transformed into a certain normal form by changing the basis. This normal form is almost diagonal in the sense that its only non-zero entries lie on the diagonal and the super-diagonal. This is made more precise in the Jordan-Chevalley decomposition. One can compare this result with the spectral theorem for normal matrices, which is a special case of the Jordan normal form. It is named in honor of Camille Jordan (1838-1922). Jordan forms of two and three dimensional linear dynamical systems may be defined according to the following theorems (Arrowsmith and Place, 1982, p. 42 and 64):

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Appendix

271

Theorem A.1. Let A be a real 2 × 2 matrix, then there is a real, nonsingular matrix M such that J = M−1 AM is one of the types:

λ1 0 (a) , λ1 > λ2 0 λ2 λ 1 (c) 0 , 0 λ0

λ0 0 (b) , 0 λ0 α −β (d) , β α

β>0

where λ0 , λ1 , λ2 , α and β are real numbers. The matrix J is said to be the Jordan form of A. Proof.

Cf. Arrowsmith and Place (1982, p. 42 and next).

Theorem A.2. Let A be a real 3 × 3 matrix, then there is a real, nonsingular matrix M such that J = M−1 AM is one of the types:  λ0 0 0  λ0 0 0

 0 0 λ1 0  , 0 λ2  1 0 λ0 0  , 0 λ1



α −β β α 0 0  λ0 1  0 λ0 0 0

 0 0 , λ1  0 1 , λ0

where λ0 , λ1 , λ2 , α and β are real numbers. The matrix J is said to be the Jordan form of A. Proof. A.4

Cf. Arrowsmith and Place (1982, p. 64 and next).

Connected region

In topology and related branches of mathematics, a connected space is a topological space which cannot be represented as the disjoint union of two or more nonempty open subsets. Connectedness is one of the principal topological properties that is used to distinguish topological spaces. A stronger notion is that of a path-connected space, which is a space where any two points can be joined by a path. A subset of a topological space X is a connected set if it is a connected space when viewed as a subspace of X. It is usually easy to think about what is not connected. A simple example would be a space consisting of two rectangles, each of which is a space and

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not adjoined to the other. The space is not connected since two rectangles are disjoint. Another good example is a space with an annulus removed. The space is not connected since you cannot connect two points, one inside the annulus and the other outside ; hence the term “connect”.

A.5

Fractal dimension

In fractal geometry, the fractal dimension, D, is a statistical quantity that gives an indication of how completely a fractal appears to fill space, as one zooms down to finer and finer scales. There are many specific definitions of fractal dimension and none of them should be treated as the universal one. From the theoretical point of view the most important are the Hausdorff dimension, the packing dimension and, more generally, the Rényi dimensions. On the other hand the box-counting dimension and correlation dimension are widely used in practice, partly due to their ease of implementation. Although for some classical fractals all these dimensions do coincide, in general they are not equivalent. It has been established (Ruelle and Takens, 1971) that a dissipative dynamical system may become chaotic starting from a dimension of the phase space greater or equal to three. This chaotic behavior with small degree of freedom is due to the sensitivity to initial conditions (S.C.I.) of trajectory curves covering strange attractor . Characteristics of such attractors are:

• phase trajectory curves are pulled towards the attractor, • two trajectory curves on the attractor diverge (S.C.I.), • dimension d of the attractor is fractal. Poincaré sections enable to understand the unpredictable feature of trajectory curves of a chaotic dynamical system confined in a bounded region of the phase space. By studying intersections of the attractor with plane sections two complementary geometric transformations are observed. The former is a stretching which moves away close trajectory curves explaining thus the so-called sensivity to initial conditions: any error is exponentially amplified as time goes by. The latter is a folding bring closer distant trajectory curves keeping the dynamics in a region of finite volume. Such mecha-

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Appendix

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273

nisms corresponds to the baker’s transformations which explains the fractal structure (infinitely folded) of strange attractors. According to Benoît Mandelbrot (1975) a set is fractal if its Hausdorff-Besicovitch dimension is not integer. However, there are many others non-integer or fractal dimensions such as Kolmogorov or capacity dimension, correlation dimension (Grassberger and Procaccia, 1983a,b), Liapounoff dimension, . . . . Some of these dimensions are defined below. A.5.1

Kolmogorov or capacity dimension

Let’s consider a set of points (e.g., a segment of length L) located in a n-dimensional space and let’s try to recover this set with (hyper)cubes of edge ε. Let N (ε) the minimal number of cubes necessary to realize such operation. Kolmogorov dimension or capacity dimension dC may be defined as the limit if it exists Ln (N (ε)) ε→0 Ln (1/ε)

dC = lim

when the edge ε of the (hyper)cubes tends to zero. A.5.2

Liapounoff exponents – Wolf, Swinney, Vastano algorithm

Let’s consider a n-dimensional iterated map xk+1 = f (xk ) k = 0, 1, 2, ... with xk ∈ Rn , f : Rn → Rn . Let J (x) be the Jacobian matrix of f , q1 f k (x) , ..., qn f k (x) , eigento their module. values of J f k (x) ordered in decreasing order ! according " Then, Liapounoff exponents n of an orbit f k (x) , k = 0, 1, 2, ... are defined by: λi = lim

N →∞

1 Ln qi f N (x) , N

i = 1, 2, ..., n

Numerical computation of Liapounoff exponents has been made possible thanks to an algorithm elaborated in the middle of the eighties by A. Wolf,

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c J. B. Swift, H. L. Swinney and J. A. Vastano (1985). A MatLab version of this algorithm made by Steve Siu and called LET, Lyapunov Exponents Toolbox. It will enable to compute Liapounoff exponents of dynamical system. Another version quite similar and called MATDS, i.e. MatLab-based c by program for Dynamical System investigation also made for MatLab Vasiliy N. Govorukhin from the Faculty of Mechanics and Mathematics, Dept. of Computational mathematics, Rostov State University. These applets are available at: http://ginoux.univ-tln.fr.

On the one hand, such estimation of Liapounoff exponents provides a qualitative and quantitative characterization of the dynamical behavior of the trajectory curves. Thus, a positive Liapounoff exponent for a dynamical system having is a signature of chaotic behavior according to Bergé et al. (1984). On the other hand, the use of Kaplan-Yorke conjecture enables to evaluate the fractal dimension of the strange attractor considered.

A.5.3

Liapounoff dimension and Kaplan-Yorke conjecture

J. Kaplan and J. A. Yorke (1979) have made the conjecture that the fractal dimension of a strange attractor may be approximated from the spectrum of the Liapounoff exponents. Such a dimension has been called Kaplan-Yorke dimension or Liapounoff dimension. A n-dimensional dynamical system has n Liapounoff exponents. The sum of these exponents measures the growth rate of n-dimensional infinitesimal volume elements: 1 dV = λ1 + λ2 + ... + λn V dt For a conservative system (Hamiltonian), this quantity is null according to Liouville’s theorem. For a dissipative system this quantity is negative and there exists an attractor towards which a set of initial conditions inside the basin of attractions are pulled towards. If the dynamical system is chaotic, at least one of the Liapounoff exponent must be positive and then a strange attractor exists in this case. By ordering Liapounoff exponents in decreasing we may conclude that λ1 must be positive for a chaotic dynamical system. Dynamical systems with more than one positive Liapounoff exponents are called “hyperchaotic”.

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275

Let’s consider the sum Sp from 1 to p of exponents where p < n, then, it is obvious that for a strange attractor there exists an integer p = j such that this sum is negative. This integer j is defined by the following conditions:

Sj =

j

λi 0 et Sj+1 =

i=1

j+1

λi < 0

i=1

The strange attractor may have a dimension contained between j and j + 1. Kaplan-Yorke conjecture is obtained by linear interpolation between dimensions j and j + 1 of volume elements. The Kaplan-Yorke dimension is defined as: j $

dKY

λi Sj i=1 =j− =j− λj+1 λj+1

Since Sj is positive and λj+1 is negative it follows that dKY > j. Numerical evaluation of dKY may become sometimes problematic, as in the case of an attractor in the shape of a bi-dimensional tore. In such case; both first Liapounoff exponents are very small and the numerical error leads to values contained anywhere between one and two. Such errors may be reduced by using a polynomial interpolation instead of a linear interpolation. That’s what propose Konstantinos E. Chlouverakis and Julian C. Sprott (2004). A.5.4

Liapounoff dimension and Chlouverakis-Sprott conjecture

While studying a three-dimensional dynamical system Chlouverakis and Sprott (2004) propose to fit Sj with a parabola. The result is: dKY =

λ2 + 3λ3 +

' 9λ22 + 6λ2 λ3 − 8λ1 λ3 + 8λ1 λ2 + λ23 2 (λ3 − λ2 )

If the dynamical system consists of ordinary differential equations (O.D.E.) and has a chaotic behavior, then λ2 must be null and the previous expression reduces to: dKY

3 1 = + 2 2

2 1−8

λ1 λ3

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For the model of Lorenz (1963), Liapounoff exponents are (0.906, 0, −14.572) the Kaplan-Yorke formulae provides for the fractal dimension the value: 2.062 while the Chlouverakis-Sprott conjecture gives the value 2.112 which is slightly greater than those of Kaplan-Yorke. According to Chlouverakis and Sprott (2004), Kaplan-Yorke conjecture is a good approximation for a strange attractor the fractal dimension of which is close to an integer and the proposed quadratic modification provides a significative difference when the fractal dimension is close to an odd halfinteger.

A.6

Identities

The aim of this section is to present definitions inherent to Differential Geometry such as the concept of n-dimensional smooth curves, generalized Frénet frame, Gram-Schmidt orthogonalization process for computing curvatures of trajectory curves in Euclidean n-space as well as proofs of identities used in this work. Within the framework of Differential Geometry, n-dimensional smooth curves, i.e. smooth curves in Euclidean n−space are defined by a regular parametric representation in terms of arc length also called natural representation or unit speed parametrization. According to Gluck (1966, 1967) local metrics properties of curvatures may be directly deduced from curves parametrized in terms of time and so natural representation is not necessary.

A.6.1

Concept of curves

(t) integral of a n-dimensional dynamical Considering trajectory curve X system (2.1) as “the motion of a variable point in a space of dimension n” leads to the following definition. Definition A.1. A smooth parametrized 1 curve in Rn is a smooth map (t) : [a, b] → Rn from a closed interval [a, b] into Rn . A map is said to be X smooth or infinitely many times differentiable if the coordinate functions = [x1 , x2 , . . . , xn ]t have continuous partial derivatives x1 , x2 , . . . , xn of X of any order. 1 with

any kind of parametrization.

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A.6.2

277

Gram-Schmidt process and Fr´ enet moving frame

There are many moving frames along a trajectory curve and most of them are not related to local metrics properties of curvatures. This is not the case for the frame of Frénet (1852). In this sub-section generalized Frénet frame for n-dimensional trajectory curves in Euclidean n-space is recalled. Let’s (t), parametrized in terms of time, is suppose that the trajectory curve X n of general type in R , i.e. that the first n − 1 time derivatives: (n−1)

˙ (t), X ¨ (t), . . . , X (t), are linearly independent for all t. X (t) of general type in Rn A moving frame along a trajectory curve X (t) forming an is a collection of i vectors u1 (t), u2 (t), . . . , ui (t) along X orthogonal basis, such that: ui (t) · uj (t) = 0

(A.1)

for all t and for i = j. Vectors ui (t) may be determined by application of the Gram-Schmidt orthogonalization process described below. (n−1)

¨ (t), . . . , X ˙ (t), X (t) be linearly Gram-Schmidt process. Let X independent vectors for all t in Rn . According to Gram-Schmidt process (Lichnerowicz (1950, p. 30), Gluck (1966)) vectors u1 (t), u2 (t), . . . , ui (t) forming an orthogonal basis are defined by: ˙ (t) u1 (t) = X

¨ (t) u1 (t) · X u1 (t) u1 (t) · u1 (t)

(t) (t) (t) · X (t) · X u u 1 2 (t) − u1 (t) − u2 (t) u3 (t) = X u1 (t) · u1 (t) u2 (t) · u2 (t)

¨ (t) − u2 (t) = X

. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . ... . .. . .. . . 

  uj (t) · X (t)  (t) − ui (t) = X   uj (t) uj (t) · uj (t) j=1 (i)

n−1

(i)

(A.2)

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Generalized Fr´ enet moving frame. Starting from the vectors u1 (t), u2 (t), . . . , ui (t) forming an orthogonal basis, generalized Frénet moving (t) of general type in Rn may be built. Thus frame for the trajectory curve X derivation with respect to time t leads to the generalized Frénet formulas in Euclidean n-space: u˙ i (t) = v

n

αij uj (t)

(A.3)

j=1

+ + + + + + + ˙ + with i = 1, 2, . . . , n and where v = +X + represents the Euclidean + = +V norm of the velocity vector field. Moreover, Eq. (A.1) implies that: u˙ i (t) · uj (t) + ui (t) · u˙ j (t) = 0

(A.4)

So, αii = 0 and αij = 0 for j < i − 1. Thus, only αi,i+1 = −αi+1,i are not identically zero. Let’s pose: κ1 = α12 ,

κ2 = α23 ,

...,

κn−1 = αn−1,n

(A.5)

The generalized Frénet formulas associated with a trajectory curve in Euclidean n-space read:  u˙ 1 (t) = vκ1 u2 (t)      u˙ (t) = v [−κ1 u1 (t) + κ2 u3 (t)]   2  u˙ 3 (t) = −vκ2 u2 (t)  ·····················     u˙ n−1 (t) = v [−κn−2 un−2 (t) + κn−1 un (t)]   ˙ un (t) = −vκn−1 un−1 (t)

(A.6)

Functions κ1 , κ2 , . . . , κn−1 are called curvatures of trajectory curve (t) of general type in Rn and κn−1 is analogous to the torsion. X (t) According to Gluck (1966, p. 702) curvatures of trajectory curves X integral of any n-dimensional dynamical systems (2.1) may be defined by: κi =

ui+1 (t) u1 (t) ui (t)

(A.7)

Since 1 ≤ i ≤ n − 1 a n-dimensional trajectory curve has (n − 1) curvatures.

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A.6.3

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279

Fr´ enet trihedron and curvatures of space curves

Fr´ enet trihedron. While normalizing the basis vectors u1 (t), u2 (t), . . . , un (t) obtained with the Gram-Schmidt process, the so-called Frénet trihedron for space curves may be deduced. Hence, it may be stated that:

u1 (t) u2 (t) u3 (t) , , u1 (t) u2 (t) u3 (t)

= τ , n, b

where τ , n and b are respectively the tangent, normal and binormal unit vectors. ˙ (t), X ¨ (t) and X (t) Let’s notice that the three first time derivatives: X represent respectively the velocity, acceleration and over-acceleration vector (t), γ (t) and γ˙ (t). Thus, from the generalized Frénet field namely: V formulas (A.6) and Gluck formulae (A.7) of curvatures, the first and second curvatures of space curves, i.e. curvature and torsion may be found again. First curvature. While replacing basis vectors u1 (t) and u2 (t) resulting from the Gram-Schmidt process in formulae (A.7), (first) curvature of space trajectory curves is given by: + + + (t)+ + +γ (t) ∧ V κ1 (t) = + +3 2 = + + u1 (t) +V + u2 (t)

Proof.

(A.8)

While using the Lagrange identity it may be established that: +2 + + ˙ ¨ + ∧X u1 2 u2 2 = +X +

So, curvature κ1 reads: + + + (t)+ + +γ (t) ∧ V κ1 (t) = + +3 2 = + + u1 (t) +V + u2 (t)

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280

Second curvature. While replacing basis vectors u1 (t), u2 (t) and u3 (t) resulting from the Gram-Schmidt process in formulae (A.7), (second) curvature, i.e. torsion of space trajectory curves is given by: ˙ (t) · γ (t) ∧ V (t) γ u3 (t) =− + κ2 (t) = +2 + u1 (t) u2 (t) (t)+ + +γ (t) ∧ V

(A.9)

+2 + + ˙ 2 2 ¨ + ∧X Proof. Still using the Lagrange identity, i.e. u1 u2 = +X + torsion κ2 reads: ˙ · γ ∧ V γ u3 (t) =− + κ2 = +2 + + u1 (t) u2 (t) +γ ∧ V +

A.6.4

First identity (n)

¨ . . . , X ˙ X, X,

˙ · =X

(n) ... ¨ X ∧ X ∧ ...∧ X

= u1 u2 . . . un (A.10)

Proof. According to Postnikov (1981, p. 215), Gram-Schmidt process can be written:

(n)

¨ . . . , X ˙ X, X,

˙ · =X

(n) ... ∧ ...∧ X ¨ ∧ X X

= u1 u2 . . . un (A.11)

Comparing Eq. (A.11) with Eq. (A.2) leads to: βii = 1

(A.12)

Using Eq. (A.11) & Eq. (A.12), inner product u1 · (u2 ∧ . . . ∧ un ) reads: ˙ · u1 · (u2 ∧ . . . ∧ un ) = X

(n)

¨ . . . , X X,

(A.13)

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281

But since Gram-Schmidt basis is orthogonal, the inner product u1 · (u2 ∧ . . . ∧ un ) may also be written: u1 · (u2 ∧ . . . ∧ un ) = u1 u2 · · · un

(A.14)

From Eq. (A.13) and Eq. (A.14) it follows that: ˙ · X

(n)

¨ . . . , X X,

= u1 u2 · · · un .

As an example, while omitting the time variable the three first GramSchmidt vectors read: Table A.1

Gram-Schmidt vec-

tors.

˙ u1 = β11 X ˙ + β22 X ¨ u2 = β21 X ... ˙ + β32 X ¨ + β33 X u3 = β31 X

Using Eq. (A.11) & Eq. (A.12), inner product u1 · (u2 ∧ u3 ) reads: ... = u1 u2 u3 ˙ · (X ¨ ∧ X) u1 · (u2 ∧ u3 ) = β11 β22 β33 X

A.6.5

Second identity Ja1 . (Ja2 ∧ . . . ∧ Jan ) = Det (J) a1 . (a2 ∧ . . . ∧ an )

Proof.

(A.15)

Equation (A.15) may also be written with inner product :

Ja1 . (Ja2 ∧ . . . ∧ Jan ) = [Ja1 , Ja2 , . . . , Jan ] = Det (Ja1 , Ja2 , . . . , Jan )

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But, since (Ja1 , Ja2 , . . . , Jan ) = J (a1 , a2 , . . . , an ) and while using determinant product property, i.e. determinant of the product is equal to the product of the determinants we have: Ja1 . (Ja2 ∧ . . . ∧ Jan ) = [Ja1 , Ja2 , . . . , Jan ] = Det (Ja1 , Ja2 , . . . , Jan ) = Det (J) Det (a1 , a2 , . . . , an ) A.6.6

Third identity

Ja1 . (a2 ∧ a3 ∧ . . . ∧ an ) + a1 . (Ja2 ∧ a3 ∧ . . . ∧ an ) + . . . + a1 . (a2 ∧ a3 ∧ . . . ∧ Jan ) = T r (J) a1 . (a2 ∧ . . . ∧ an ) Proof.

(A.16)

The proof is based on inner product properties.

To the functional jacobian matrix J is associated an eigenbasis: Yλ1 , Yλ2 , . . . , Yλn Let suppose that there exists a transformation2 such that: to each vector ai corresponds the eigenvector Yλi with i = 1, . . . , n. Then, each inner product of the left hand side equation (A.16) may be transformed into Ja1 · (a2 ∧ a3 ∧ . . . ∧ an ) = λ1a1 · (a2 ∧ a3 ∧ . . . ∧ an ) = λ1a1 · (a2 ∧ a3 ∧ . . . ∧ an ) a1 · (Ja2 ∧ a3 ∧ . . . ∧ an ) = a1 · (λ2a2 ∧ a3 ∧ . . . ∧ an ) = λ2a1 · (a2 ∧ a3 ∧ . . . ∧ an ) ................................................... a1 · (a2 ∧ a3 ∧ . . . ∧ Jan ) = a1 · (a2 ∧ a3 ∧ . . . ∧ λnan ) = λna1 · (a2 ∧ a3 ∧ . . . ∧ an ) Making the sum of these factors the proof is stated.

2 By considering that each vector ai may be spanned on the eigenbasis calculus is longer and tedious but leads to the same result.

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Appendix

A.7 A.7.1

DGeometry

283

Homeomorphism and diffeomorphism Homeomorphism

A function f between two topological spaces X and Y is called a homeomorphism if it has the following properties: • f is a bijection • f is continuous, • the inverse function f −1 is continuous (f is an open mapping). A.7.2

Diffeomorphism

A diffeomorphism is a map between manifolds which is differentiable and has a differentiable inverse.

A.8 A.8.1

Differential equations Two-dimensional dynamical systems

According to D’Alembert theorem, any single 2nd order differential equation may be transformed into a system of two simultaneous first-order equations which reads in a symmetric form: dx dt dy dt

= P (x, y) = Q (x, y)

⇔

dy dx = = dt P (x, y) Q (x, y)

This symmetric form may also be written as a first order and first degree differential equation involving two variables, namely x and y which represents a differential 1-form.

Q (x, y) dx − P (x, y) dy = 0 = φ (x, y) = C which The general integral of this system reads: φ(X) defines a family of curves depending on an arbitrary constant C.

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284

A.8.2

Three-dimensional dynamical systems

According to D’Alembert theorem, any single 3rd order differential equation may be transformed into a system of three simultaneous first-order equations which reads in a symmetric form:  dx  dt = P (x, y, z) dy dt = Q (x, y, z)  dz dt = R (x, y, z)

⇔

dy dz dx = = = dt P (x, y, z) Q (x, y, z) R (x, y, z)

This symmetric form leads to two first order and first degree differential equations involving three variables, namely x, y and z, i.e. to two differential 1-form.

P1 (x, y, z) dx + Q1 (x, y, z) dy + R1 (x, y, z) dz = 0 P2 (x, y, z) dx + Q2 (x, y, z) dy + R2 (x, y, z) dz = 0

= φ1 (x, y, z) = C1 , The general integral of this system reads: φ1 (X) φ2 (X) = φ2 (x, y, z) = C2 which defines a family of curves depending on two arbitrary constants C1 and C2 . Historical notes In a letter to Leibniz (1647-1716) through the intermediary of Olenburg dated the 26th October 1676, Newton (1642-1727) formulated the first general statement of the problem of integrating a differential equation in the following anagram: 6a, 2c, d, ae, 13e, 2f, 7i, 3l, 9n, 4o, 4q, 2r, 4s, 8t, 12v, x, which was deciphered thus: Data aequatione quotcumque fluenes quantitates involvente, fluxiones invenire et vice versa 3 . Thus, Leibniz discovered the method of separation of variables and introduced the modern differential and integration notations. While in the Philosophiæ Naturalis Principia Mathematica published in July 5th 1687, Newton stated the laws of motion forming the foundation of classical mechanics. 3 Given any equation, involving fluent quantities (integrals), to find the fluxions (differentials), and vice versa.

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A.9

285

Generalized Tangent Linear System Approximation

A.9.1

Assumptions

The generalized tangent linear system approximation requires that the dynamical system (2.1) satisfies the following assumptions: X) defined in E are (H1 ) The components fi , of the velocity vector field ( ∞ continuous, C functions in E and with values included in R. (H2 ) The dynamical system (2.1) satisfies the nonlinear part condition (Rossetto et al., 1998), i.e. that the influence of the nonlinear part X) of this system is of the Taylor series of the velocity vector field ( overshadowed by the fast dynamics of the linear part. X) d ( −X 0 )2 (X) = (X0 ) + (X − X0 ) + O (X dX

(A.17)

X0

(H3 ) The functional jacobian matrix associated to dynamical system (2.1) has at least a “fast” eigenvalue λ1 , i.e. with the largest absolute value of the real part. A.9.2

Corollaries

To the dynamical system (2.1) is associated a tangent linear system defined as follows: dδ X 0 )δ X = J(X dt where

=X −X 0, δX

(A.18)

X) d ( 0 = X (t0 ) and 0) X = J(X dX

X0

Corollary A.1. The nonlinear part condition implies that the velocity varies slowly in the vicinity of the slow manifold. This involves that the functional jacobian 0 ) varies slowly with time, i.e. J(X dJ =0 dt X 0

(A.19)

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The solution of the tangent linear system (A.18) is written: = eJ(X 0 )(t−t0 ) δ X (t0 ) δX

(A.20)

So, = δX

n

ai Yλi

(A.21)

i=1

where n is the dimension of the eigenspace, ai represents coefficients depending explicitly on the co-ordinates of space and implicitly on time and Yλi the eigenvectors associated in the functional jacobian of the tangent linear system. Corollary A.2. In the vicinity of the slow manifold the velocity of dynamical system (2.1) and that of the tangent linear system (A.18) merge. dδ X = VT ≈ V dt

(A.22)

T represents the velocity vector associated with the tangent linear where V system. The tangent linear system approximation consists of spreading the velocity vector field V on the eigenbasis associated to the functional jacobian matrix of the tangent linear system. While taking account of (A.18) and (A.21) we have according to (A.22): n dδ X 0 )δ X = J(X 0) = J(X ai Yλi dt i=1

=

n

0 )Yλi = ai J(X

i=1

n

ai λi Yλi

(A.23)

i=1

Thus, Corollary 2 provides: dδ X = = VT ≈ V ai λi Yλi dt i=1 n

(A.24)

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Then, existence of an evanescent mode in the vicinity of the slow manifold implies according to theorem Tikhonov (1948, 1952) that a1 λ1 1. So, the coplanarity condition (A.24) provides the slow manifold equation of a n-dimensional dynamical system (2.1). Proposition A.1. The coplanarity condition between the velocity vector of a n-dimensional dynamical system and the slow eigenvectors Yλi field V associated to the slow eigenvalues λi of its functional jacobian provides the slow manifold equation of such system. = V

n

ai Yλi = a2 Yλ2 + . . . + an Yλn

i=2

⇔ = V . Yλ2 ∧ . . . ∧ Yλn = 0 φ(X)

(A.25)

An alternative proposed by (Rossetto et al., 1998) uses the “fast” eigenvector on the left associated with the “fast” eigenvalue of the transposed functional jacobian of the tangent linear system. In this case the velocity is then orthogonal with the “fast” eigenvector on the left. vector field V This orthogonality condition also provides the slow manifold equation of a n-dimensional dynamical system (2.1). Proposition A.2. The orthogonality condition between the velocity vector of a n-dimensional dynamical system and the fast eigenvector t Yλ1 field V on the left associated with the fast eigenvalue λ1 of its transposed functional jacobian provides the slow manifold equation of such system. = V · t Yλ1 = 0 φ(X)

(A.26)

Proposition A.3. Both coplanarity and orthogonality conditions providing the slow manifold equation are equivalent. Proof.

While using the following identity the proof is obvious:

Yλ2 ∧ Yλ3 ∧ . . . ∧ Yλn = t Yλ1

(A.27)

So, coplanarity and orthogonality conditions are completely equivalent.

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Since for low-dimensional two and three dynamical systems the proof has been already established (Ginoux and Rossetto, 2006) while using the Tangent Linear System Approximation, for high-dimensional dynamical systems it may be deduced from its generalization presented above. Thus, according to the generalization of the Tangent Linear System Approximation the slow manifold equation of a n-dimensional dynamical system may be written: =V . Yλ2 ∧ . . . ∧ Yλn = 0 φ(X)

⇔

V =

n

ai Yλi

(A.28)

i=2

In the framework of the Generalized Tangent Linear System Approximation the functional jacobian matrix associated to the dynamical system is supposed to be locally stationary: dJ =0 dt

(A.29)

As a consequence, time derivatives of acceleration vectors reads: (n)

γ = J (n+1) V = J (n)γ

Then, mapping the flow of the tangent linear system, i.e. functional jacobian operator J to the velocity vector field spanned on the eigenbasis (A.28) leads to: = γ = JV

n

ai J Yλi = a2 J Yλ2 + . . . + an J Yλn

i=2

...................................................

= J (n−2) V

(n−2)

γ

=

n i=2

ai J (n−2) Yλi = a2 J (n−2) Yλ2 + . . . + an J (n−2) Yλn

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289

By using the eigenequation: J Yλk = λk Yλk these equations may be written: = γ = JV

n

ai λi Yλi = a2 λ2 Yλ2 + . . . + an λn Yλn

i=2

...................................................

= J (n−2) V

(n−2)

γ

=

n

Yλ2 + . . . + an λn−2 Yλn ai J (n−2) Yλi = a2 λn−2 n 2

i=2

Under the assumptions of the Tangent Linear System Approximation, it (n−2)

is obvious that the vectors V ,γ , . . . , γ spanned on the same eigenbasis Yλ2 , Yλ3 , . . . , Yλn are “hypercoplanar”. This implies that

(n−2)

· γ ∧ γ˙ ∧ . . . ∧ γ V

=0

⇔

˙ · X

(n) ... ¨ X ∧ X ∧ ... ∧ X

=0

Thus, it has been stated that the Flow Curvature Method generalizes and encompasses the Tangent Linear System Approximation.

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Mathematica Files

Part I Dynamical Systems Chapter 1 Differential Equations There are no Mathematica Files in Chapter 1.

Chapter 2 Dynamical Systems • MF 01 Free Fall • MF 02 Predator-prey model • MF 03 Inverted pendulum • MF 04 Saddle Focus • MF 05 First Limit cycle • MF 06 Unforced Duffing oscillator • MF 07 Lorenz Butterfly • MF 08 Spherical Pendulum • MF 09 Hnon - Heiles Hamiltonian

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Chapter 3 Invariant Sets • MF 10 Unstable Manifold Chapter 4 Local Bifurcations • MF 11 Nonuniqueness of the Centre Manifold Chapter 5 Slow-Fast Dynamical Systems • MF 12 Geometric Singular Perturbation Method 2D - Van der Pol System • MF 13 Geometric Singular Perturbation Method 3D - Chua’s System Chapter 6 Integrability • MF 14 Integrable System • MF 15 General Integral • MF 16 General Integral of Homogeneous Dynamical Systems • MF 17 Multiplierof Homogeneous Dynamical Systems • MF 18 First Integral of Affine and Projective models • MF 19 Kapteyn-Bautin system

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Part II Differential Geometry

Chapter 7 Differential Geometry There are no Mathematica Files in Chapter 7.

Chapter 8 Dynamical Systems • MF 20 Unforced Duffing oscillator • MF 21 Lorenz Model

Chapter 9 Invariant Sets • MF 22 Local Invariance • MF 23 Volterra-Lotka predator-prey model 2D • MF 24 Volterra-Lotka predator-prey model 3D • MF 25 Nondecomposable quadrics - (PRT) model

Chapter 10 Local Bifurcations • MF 26 Centre Manifold Approximation 2D • MF 27 Centre Manifold Approximation 3D • MF 28 Centre Manifold depending upon a parameter 2D - Duffing system • MF 29 Centre Manifold depending upon a parameter 3D - Chen’s system

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Chapter 11 Slow-Fast Dynamical Systems • MF 30 Flow Curvature Method 2D - Van der Pol system • MF 31 Flow Curvature Method 3D - Chua’s system • MF 32 Non-singularly perturbed systems: Lorenz Model

Chapter 12 Integrability • MF 33 Van der Pol system • MF 34 First integral of Volterra-Lotka system • MF 35 First integral of quadratic system

Chapter 13 Inverse Problem • MF 36 Inverse Problem 2D - Van der Pol system • MF 37 Inverse Problem 3D - Lorenz model

Part III Applications Chapter 14 Dynamical Systems • MF 38 FitzHugh-Nagumo model 2D • MF 39 Pikovskii-Rabinovich-Trakhtengerts model 3D

Chapter 15 Invariant Sets - Integrability • MF 40 Pikovskii-Rabinovich-Trakhtengerts (PRT) • MF 41 Rikitake model

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• MF 42 Chua’s model • MF 43 Lorenz model

Chapter 16 Local Bifurcations • MF 44 Chua’s model • MF 45 Lorenz model Chapter 17 Slow-Fast Dynamical Systems Piecewise linear models 2D & 3D • MF 46 Van der Pol piecewise linear model 2D • MF 47 Chua’s piecewise linear model 3D Singularly perturbed systems 2D & 3D • MF 48 FitzHugh-Nagumo model 2D • MF 48 Chua’s cubic model 3D Slow-fast dynamical 2D & 3D • MF 50 Brusselator model 2D • MF 51 Pikovskii-Rabinovich-Trakhtengerts model 3D • MF 52 Rikitake model 3D Piecewise linear models 4D & 5D • MF 53 Chua’s piecewise linear model 4D • MF 54 Chua’s piecewise linear model 5D

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Singularly perturbed systems 4D & 5D • MF 55 Chua’s cubic model 4D • MF 56 Chua’s cubic model 4D • MF 57 Hompolar Dynamo Model 4D • MF 58 Mofatt model 5D • MF 59 Magnetoconvection model 5D Non-Autonomous Dynamical Systems • MF 60 Forced Van der Pol

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Bibliography

Abraham, R. H. & Christopher D. S. (1982-88). Dynamics: The Geometry of Behavior, Part 1-4, Vismath Volume I-IV, (Aerial Press, Inc.) Abraham, R. H. & Marsden, J. E. (1987) Foundations of Mechanics, (Second Edition, Addison-Wesley Publishing Company, Inc., Redwood City, CA), http://caltechbook.library.caltech.edu/103/. Algaba, A., Freire, E., Gamero, E. & Rodriguez-Luis, A. J. (1999). On the TakensBogdanov bifurcation in the Chua’s equation, IEICE Transactions on Fundamentals of Electronics Communications & Computer Sciences, vol.E82A, no.9, pp. 1722–1728. Andronov, A. A., Khaikin, S. E. & Vitt, A. A. (1937). Theory of oscillators, I, Moscow (Engl. transl., Princeton Univ. Press, Princeton, N. J., 1949). Arnold, V. I. (1963). Small denominators and problems of stability of motion in classical and celestial mechanics, Russian Math. Surveys 18(6), pp. 85–191. ´ Arnold, V. I. (1974). Equations différentielles ordinaires, Mir, Moscow. Arnold, V. I. (1978). Mathematical methods of classical mechanics, SpringerVerlag, New-York. Arnold, V. I. (1984). Chapitres supplémentaires de la théorie des équations différentielles ordinaires, Mir, Moscow. Arnold, V. I. (1996). Remarks on the extactic points of plane curves, The Gelfand Mathematical Seminars (1993-1995) Birkhuser, Boston, pp. 1122. Arrowsmith, D. K. & Place, C.M. (1982). Ordinary Differential Equations, Chapman & Hall. Arrowsmith, D. K. & Place, C.M. (1990). An Introduction to Dynamical Systems, Chapman & Hall. Autonne, L. (1891). Sur la théorie des équations différentielles du premier ordre ´ et du premier degré, J. Ecole Polytechnique 1ère et 2ème partie C. LXI, pp. 35–122. Autonne, L. (1892). Sur la théorie des équations différentielles du premier ordre ´ et du premier degré, J. Ecole Polytechnique 1ère et 2ème C. LXI, 3ème partie C. LXII, pp. 47–180. Bautin, N. N. (1954). On the number of limit cycles which appear with the variation of coefficients from an equilibrium position of focus or center type,

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Index

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mali conviventi, Mem. Acad. Lincei, III, 6, pp. 31–113. Abstract in Volterra, V. (1926). Fluctuations in the abundance of a species considered mathematically, Nature, vol. CXVIII, pp. 558-560. Volterra, V. (1928). Variations and fluctuations of the number of individuals in animal species living together, Journal du Conseil international pour l’exploration de la mer, Copenhague, vol. III, 1, pp. 3-51. Volterra, V. (1931). Le¸cons sur la Th´ eorie Mathématique de la Lutte pour la Vie, Gauthier-Villars, Paris. Wasow, W. R. (1965). Asymptotic Expansions for Ordinary Differential Equations, Wiley-Interscience, New York. Weil, J. A. (1995). Constantes et polynˆ omes de Darboux en algèbre différentielle : ´ application aux syst` emes différentiels linéaires , Ecole Polytechnique, Ph-D (Math). Wiggins, S. (1990). Introduction to Applied Nonlinear Dynamical Systems and Chaos, Springer-Verlag, New York. Wolf, A., Swift, J. B., Swinney, H. L. & Vastano, J. A. (1985). Determining Lyapunov Exponents from a Time Series, Physica D, Vol. 16, pp. 285–317.

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Index

attractor, 31, 32, 263, 267, 272, 274, 275 attractor structure, 267 autonomous, xi, xii, 7, 8, 12, 47, 261, 264, 266

curvature, vii, ix, 123, 125, 127–129, 131, 132, 152–154, 184, 195, 200, 207, 265, 267, 276–280 curvature of the flow, vii curve, vii, 11, 24–26, 29, 61, 86, 92, 93, 95, 124, 127–132, 136, 152, 276, 278, 279, 283, 284

Bendixson, 29–31 bifurcation, 47, 48, 53, 57, 62, 64–66, 123, 133, 159, 174, 181, 239, 240, 292, 293, 295 Brusselator, xi, 228, 248, 295

Darboux, viii, x, 87, 92, 94, 96, 97, 99, 102, 104, 106, 107, 109, 110, 114, 117, 118, 145, 146, 157, 186, 187, 190, 203, 208–212, 230, 266 Darboux invariance theorem, ix, x, 96, 97, 145, 146, 184, 186, 190, 242, 244, 246–250, 252, 254, 256, 257, 259, 260, 262, 264–266 diffeomorphism, 13, 283 differential equation, vii, 3–8, 11, 28, 31, 47, 50, 53, 60, 66, 85, 86, 88–94, 123, 135, 159, 167, 184, 228, 231, 232, 251, 253, 255, 257, 283, 284 differential geometry, vii, ix, xi, xii, 121, 123, 127, 152, 184, 265, 276, 293 dissipative, 31–33, 94, 272, 274 divergence, 32, 86, 90 Duffing, 30, 141, 142, 169, 171, 291, 293 dynamical system, vii, viii, x, 1, 6–14, 17, 18, 21–27, 29, 31, 32, 34, 35, 41–44, 46–51, 53–55, 57, 59–61, 63, 64, 66, 67, 70, 81–83, 85, 86, 88–90,

Cauchy, vii, 8, 9, 24, 31, 48, 81, 86, 89, 92, 123, 124, 135, 159, 167, 185, 215, 217 centre manifold, ix, xi, 41, 46, 48–54, 57, 60, 61, 74, 123, 133, 159–164, 166–168, 170, 171, 174, 175, 181, 237, 239, 240, 265, 267, 292, 293 chaotic attractor, 249 Chua, viii, xi, 22, 71, 72, 80, 81, 198, 228, 232, 233, 237, 243–245, 247, 251–258, 267, 292, 294–296 Chua’s invariant hyperplanes, 253, 255 Chua’s slow invariant manifold, 199, 247 codimension, 53, 60, 133, 159, 181, 240 complex dynamics, 71 conservative, 32, 35, 94, 274 309

DGeometry

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310



92–94, 96–108, 112, 113, 123, 124, 127–135, 137–139, 141, 143–147, 153–155, 157, 159, 160, 162, 164, 166–169, 171, 175–177, 180, 183–186, 188, 191, 195, 196, 200, 203, 204, 206, 207, 209, 211, 212, 215–219, 222–224, 238, 239, 265–267, 269, 276, 278, 285–288, 291–294 Fenichel, x, 69, 70, 72–74, 77, 80, 83, 183, 188, 190, 266 first integral, viii–xi, 34–36, 93–95, 97–99, 102–104, 108–111, 113, 116–119, 145, 146, 154, 203–214, 230, 232, 233, 265, 266, 292, 294 FitzHugh-Nagumo, xi, 227, 245, 246, 294, 295 FitzHugh-Nagumo slow invariant manifold, 246 fixed point, ix, xi, 9–15, 17–19, 21, 24–31, 42–45, 48, 50, 60, 62–66, 102, 108, 123, 124, 128, 130, 131, 133, 135–144, 188, 222, 224, 227, 228, 244, 245, 252, 254, 267 fixed point stability, ix, 265 flow curvature manifold, vii, x, 123, 133–141, 144, 145, 148–151, 154, 155, 157, 159–164, 166–168, 170, 171, 175, 176, 181, 185, 187, 189, 191, 194–196, 199, 200, 206–208, 210–213, 215–219, 221–224, 227–231, 233, 234, 237, 242, 244, 246–252, 254, 256, 257, 259, 260, 262, 264–267 flow curvature method, vii, ix, xii, 82, 102, 123, 133, 157, 188, 189, 193, 195, 196, 198–200, 205, 264–268, 289, 294 Forced Van der Pol, xi, 263, 264, 266, 296 Galois, 144, 184 Geometric Singular Perturbation Theory, viii, x, 46, 69, 70, 72, 74, 82, 83, 183, 188, 189, 191, 193, 194,

196, 198–200, 205, 266, 292 Grobman, 13, 45 Groebner, 223, 224 Hénon-Heiles Hamiltonian, 38, 39 Hnon-Heiles Hamiltonian, 291 Hamiltonian, viii, 34, 35, 37–39, 203, 274 harmonic oscillator, 34 Hartman, 13, 45 homeomorphism, 13, 283 homopolar dynamo, xi, 258, 259 Hopf, 67, 159 Hopf bifurcation, 66, 67 hyperbolic, 13, 25, 44, 45, 69, 73, 183 hyperbolic points, 37 implicit function theorem, 42, 73, 76, 77, 79, 80, 189 integrability, 41, 85–93, 95, 96, 123, 145, 157, 203, 209, 265, 266, 292, 294 invariant manifold, 41, 44, 46, 48, 61, 69, 97, 98, 100–103, 105, 107, 108, 123, 133, 145–148, 150–157, 183, 187, 203, 206, 233, 234 invariant tori, 37 inverse problem, x, 134, 215, 221, 222, 266, 267, 294 inverted pendulum, 17, 22 Jacobian, 13, 14, 17, 18, 21, 22, 24–26, 41–43, 45, 48, 49, 60–62, 64–66, 82, 83, 88, 125, 137–139, 142, 143, 160, 184–189, 244, 285–288 Jordan fom, 270, 271 Jordan form, 15, 19, 25, 270 K.A.M. theorem, viii, 37 K.A.M. tori, 37, 39 Kapteyn-Bautin, 119, 213, 214, 292 LaSalle, 41 Liapounoff, viii, 13, 14 Liapounoff dimension, 273–275

DGeometry

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Index

Liapounoff exponents, 273–276 Lie derivative, viii, x, 26, 34, 87, 112, 137–139, 146, 147, 150–152, 155, 156, 187, 190, 195, 200, 204, 205, 230, 235, 242, 244, 248–250, 252, 256, 257, 259, 260, 262, 264, 266, 269 limit cycle, viii, 26–29, 31, 67, 291 linear, 13, 22, 43, 55, 56, 130, 133, 145, 148–150, 152–154, 157, 176, 184–186, 206, 243, 251–253, 255, 257, 270, 285–288 linear invariant manifold, ix, 151, 154, 207–213, 229, 265 Liouville, 32, 85, 203, 274 local bifurcations, 265 Local first integral, 205 Local invariance, 148 Lorenz, x, xi, 33, 82, 83, 144, 200, 201, 217, 224, 234, 239, 240, 261, 266, 276, 293–295 Lorenz butterfly, 33, 291 Lorenz slow manifold, 201 magnetoconvection, xi, 261, 262, 296 Mofatt, xi, 258, 260, 296 non-autonomous, 241, 263 non-singularly perturbed systems, x, 81, 83, 200, 266, 294 nonlinear, 22 nonlinear invariant manifold, 155, 230, 265 normal forms, ix, x, 54, 55, 57, 59, 123, 133, 159, 175, 176, 179, 180, 229, 265, 270 null-clines, 9, 11, 12 orbit, 13, 30, 37, 42, 69, 183, 273 osculating plane, ix, 127–132, 137–141, 144, 207, 244 pendulum, 3, 4, 152 phase, 7, 8, 10, 11, 14, 16, 17, 29, 32, 34, 47, 82, 94, 135, 242, 244, 253, 255, 256, 258, 259, 261, 262, 264,

DGeometry

311

267, 272 piecewise linear model, xi, 72, 241–245, 251–254, 295 Pikovskii, xi, 228, 229, 249, 294, 295 pitchfork bifurcation, viii, 54, 62, 64–66, 159, 174, 181, 239, 240 Poincaré, ix, 27–29, 31 Poincaré index, viii Poincaré section, 39, 272 Poincar, 85, 124, 141 quadratic system, 211 relaxation oscillations, 11, 23, 241 relaxation oscillator, 71 Rikitake, xi, 231, 250, 258, 266, 294, 295 saddle, 15, 17, 19, 21, 25, 30, 63, 139, 141, 142, 159, 181 saddle points, 37 saddle-focus, 142 saddle-node, 142 saddle-node bifurcation, 62, 63 singular approximation, 23, 24, 77, 80, 82, 193, 195, 198, 204, 205, 246–248 Singular Perturbation Method, 72, 188, 195, 199, 247, 292 singularly perturbed systems, viii, x, xi, 12, 23, 24, 46, 69–72, 74, 75, 78, 81–83, 134, 145, 183, 184, 188, 191, 192, 196, 197, 200, 204, 241, 245, 247, 249, 250, 255, 259, 260, 262, 266, 295, 296 slow invariant manifold, viii–xi, 69, 73, 74, 77, 80, 82, 83, 134, 183, 185, 188–192, 194, 196, 197, 199, 229, 242, 244, 247–251, 254, 256–266 slow-fast dynamical systems, viii, 24, 81, 184, 187, 188, 195, 248, 266, 292, 294, 295 spherical pendulum, 35, 36, 204, 291 stability, 13, 14, 16, 20, 47, 62, 64, 65, 123, 133, 135, 137, 139, 141, 144, 145, 183, 227

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strange attractor, viii, 32, 35, 272–276 tangent linear system approximation, xi, 184–186, 285, 286, 288, 289 torsion, 127, 131, 132, 152, 184, 200, 278–280 trajectory curve, vii, ix, 8–10, 26, 29, 31, 32, 34–38, 94, 123–128, 132, 133, 153, 154, 184, 206, 207, 265, 267, 268, 272, 274, 276–280 transcritical, 159, 171, 181 transcritical bifurcation, 62–64 Van der Pol, viii, xi, 11, 23, 26–28, 193, 195, 204, 205, 216, 220, 223, 227, 241, 242, 245, 263, 264, 292, 294, 295 Volterra, 11, 12, 22, 112, 151, 154, 208, 209, 293, 294

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