Modeling and Computations in Dynamical S: Dedicated to John Von Neumann

WORLD SCIENTIFIC SERIES ON

ries Series Editor: Leon 0 . Chua

MODELING AND COMPUTATIONS IN DYNAMICAL SYSTEMS In commemoration of the 100th anniversary of the birth of John von Neumann edited by EUSEBIUS J DOEDEL, CABOR DOMOKOS & IOANIMIS G KEVREKIDIS

World Scientific

MODELING AND COMPUTATIONS IN DYNAMICAL SYSTEMS In commemoration of the 100th anniversary of the birth of John von Neumann

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

SPECIAL THEME ISSUES AND PROCEEDINGS

Volume 1:

Chua's Circuit: A Paradigm for Chaos Edited by R. N. Madan

Volume 2:

Complexity and Chaos Edited by N. B. Abraham, A. M. Albano, A. Passamante, P. E. Rapp, and R. Gilmore

Volume 3:

New Trends in Pattern Formation in Active Nonlinear Media Edited by V. Perez-Villar, V. Perez-Munuzuri, C. Perez Garcia, and V. I. Krinsky

Volume 4:

Chaos and Nonlinear Mechanics Edited by T. Kapitaniak and J. Brindley

Volume 5:

Fluid Physics — Lecture Notes of Summer Schools Edited by M. G. Velarde and C. I. Christov

Volume 6:

Dynamics of Nonlinear and Disordered Systems Edited by G. Martfnez-Mekler and T. H. Seligman

Volume 7:

Chaos in Mesoscopic Systems Edited by H. A. Cerdeira and G. Casati

Volume 8:

Thirty Years After SharkovskiT's Theorem: New Perspectives Edited by L Alseda, F. Balibrea, J. Llibre, and M. Misiurewicz

Volume 9:

Discretely-Coupled Dynamical Systems Edited by V. Perez-Munuzuri, V. Perez-Villar, L. O. Chua, and M. Markus

Volume 10:

Nonlinear Dynamics & Chaos Edited by S. Kim, R. P. Behringer, H.-T. Moon, and Y. Kuramoto

Volume 11: Chaos in Circuits and Systems Edited by G. Chen and T. Ueta Volume 12: Dynamics and Bifurcation of Patterns in Dissipative Systems Edited by G. Dangelmayr and I. Oprea

& I WOBLD SCIENTIFIC SERIES ON * • * >

NONLINEAR SCIENCE Series Editor: Leon 0. Chua

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In commemoration of the 100th anniversary of the birth of John von Neumann

edited by

Eusebius J. Doedel Concordia university, Canada

Gabor Domokos Budapest university of Technology and Economics, Hungary

loannis G. Kevrekidis Princeton university, USA

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MODELING AND COMPUTATIONS IN DYNAMICAL SYSTEMS

NEWJERSEY

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• SINGAPORE

• BEIJING

• SHANGHAI

• HONGKONG

• TAIPEI

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CHENNAI

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.

Cover Illustration: The image is an artistic rendering by Greg Jones (University of Bristol) of the Lorenz manifold as computed by the five different methods; see the chapter "A Survey of Methods for Computing (Un)Stable Manifolds of Vector Fields", by B. Krauskopf, H. M. Osinga, E. J. Doedel, M. E. Henderson, J. Guckenheimer, A. Vladimirsky, M. Dellnitz and O. Junge.

MODELING AND COMPUTATIONS IN DYNAMICAL SYSTEMS Copyright © 2006 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

981-256-596-5

Typeset by Stallion Press E-mail: [email protected]

Printed bv Fulsland Offset Printins (SVPte Ltd, Singapore

CONTENTS Editorial

1

Transport in Dynamical Astronomy and Multibody Problems M, Dellnitz, 0. Junge, W. S. Koon, F. Lekien, M. W. bo, J. E. Marsden, K. Padberg, R. Preis, S. D. Ross and B. Thiere

3

A Brief Survey on the Numerical Dynamics for Functional Differential Equations B. M. Garay

33

Bifurcations and Continuous Transitions of Attractors in Autonomous and Nonautonomous Systems P. E. Kloeden and S. Siegmund

47

A Survey of Methods for Computing (Un)Stable Manifolds of Vector Fields B. Krauskopf, H. M. Osinga, E. J. Doedel, M. E. Henderson, J. Guckenheimer, A. Vladimirsky, M. Dellnitz and 0. Junge

67

Commutators of Skew-Symmetric Matrices A. M. Bloch and A. Iserles

97

Simple Neural Networks that Optimize Decisions E. Brown, J. Gao, P. Holmes, R. Bogacz, M. Gilzenrat and J. D. Cohen

107

Newton Flow and Interior Point Methods in Linear Programming J.-P. Dedieu and M. Shub

131

Numerical Continuation of Branch Points of Equilibria and Periodic Orbits E. J. Doedel, W. Govaerts, Yu. A. Kuznetsov and A- Dhooge

145

Coarse-Grained Observation of Discretized Maps G. Domokos

165

Multiple Helical Perversions of Finite, Intristically Curved Rods G. Domokos and T. J. Healey

175

Bifurcations of Stable Sets in Noninvertible Planar Maps J. P. England, B. Krauskopf and H. M. Osinga

195

Multiparametric Bifurcations m an Enzyme-Catalyzed Reaction Model E. Freire, b. Pizarro, A. J. Rodriguez-buis and F. Fernandez-Sanchez

209

v

Straightforward Computation of Spatial Equilibria of Geometrically Exact Cosserat Rods T. J. Healey and P. G. Mehta Multiparameter Parallel Search Branch Switching M. E. Henderson

253

271

Equation-Free, Effective Computation for Discrete Systems: A Time Stepper Based Approach J. Moiler, 0. Runborg, P. G. Kevrekidis, K. Lust and I. G. Kevrekidis

279

Model Reduction for Fluids, Using Balanced Proper Orthogonal Decomposition C. W. Rowley

301

Bifurcation Tracking Algorithms and Software for Large Scale Applications A. G. Salinger, E. A. Burroughs, R. P. Pawlowski, E. T. Phipps and L. A. Romero An Algorithm for Finding Invariant Algebraic Curves of a Given Degree for Polynomial Planar Vector Fields G. Swirszcz

319

337

EDITORIAL The papers in this issue are based on lectures presented at the October 2003 Budapest workshop on Modeling and Computations in Dynamical Systems, and complemented by selected additional contributions. The workshop, organized by G. Domokos, was held in commemoration of the 100th anniversary of the date of birth of John von Neumann, and made possible by generous support from The Thomas Cholnoky Foundation. Von Neumann made fundamental contributions to Computing, and he had a keen interest in Dynamical Systems, specifically, Hydrodynamic Turbulence. It was especially appropriate therefore, to dedicate the workshop (and this special issue) to the memory of von Neumann, one of the greatest and most influential mathematicians of the 20th century. While the topic of the Budapest workshop was rather well-defined, concentrating on modeling and computations in dynamical systems, the gathering attracted a diverse group of prominent researchers, theoreticians as well as computational scientists, with fields of expertise ranging from numerical techniques, including large scale computing, to fundamental aspects of dynamical systems. The papers in this special issue reflect these diverse interests, and, in fact, the wide-ranging nature of the field of Dynamical Systems. Applications of the work reported in this special issue include geometric integration, neural networks, linear programming, dynamical astronomy, chemical reaction models, and structural and fluid mechanics. Busebius Doedel, Concordia University, Montreal, Canada Gabor Domokos, Budapest University of Technology and Economics, Hungary Ioannis Kevrekidis, Princeton University, USA

1

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T R A N S P O R T IN DYNAMICAL A S T R O N O M Y AND MULTIBODY PROBLEMS MICHAEL DELLNITZ*, OLIVER JUNGE*, WANG SANG K O O N t , F R A N C O I S LEKIEN*, MARTIN W. LO § , J E R R O L D E. MARSDEN^, K A T H R I N PADBERG*, R O B E R T PREIS*, SHANE D. ROSS*, and BIANCA THIERE* *Faculty of Computer Science, Electrical Engineering and Mathematics, University of Paderborn, D-33095 Paderborn, Germany ^Control and Dynamical Systems, MC 107-81, California Institute of Technology, Pasadena, CA 91125,

USA

^•Department of Mechanical and Aerospace Engineering, Princeton University Engineering Quad, Olden Street, Princeton, NJ 08544-5263, USA ^Navigation and Mission Design, Jet Propulsion Laboratory, California Institute of Technology, M/S 301-140L, 4800 Oak Grove Drive, Pasadena, CA 91109, USA Received April 28, 2004; Revised July 5, 2004

We combine the techniques of almost invariant sets (using tree structured box elimination and graph partitioning algorithms) with invariant manifold and lobe dynamics techniques. The result is a new computational technique for computing key dynamical features, including almost invariant sets, resonance regions as well as transport rates and bottlenecks between regions in dynamical systems. This methodology can be applied to a variety of multibody problems, including those in molecular modeling, chemical reaction rates and dynamical astronomy. In this paper we focus on problems in dynamical astronomy to illustrate the power of the combination of these different numerical tools and their applicability. In particular, we compute transport rates between two resonance regions for the three-body system consisting of the Sun, Jupiter and a third body (such as an asteroid). These resonance regions are appropriate for certain comets and asteroids. Keywords: Three-body problem; transport rates; dynamical systems; almost invariant sets; graph partitioning; set-oriented methods; invariant manifolds; lobe dynamics.

Contents

1. Introduction 1.1. Need for modification of current transport calculations 1.1.1. Chemistry 1.1.2. Dynamical astronomy 1.2. Current methods for the study of transport in the PCR3BP 1.2.1. Analytical methods: single resonance theory and resonance overlap criterion 3

4 6 6 6 6 6

4

2.

3.

4.

5.

M. Dellnitz et al.

1.2.2. Toward a global picture of the phase space 1.2.3. Mars escape rates 1.3. Set oriented approach to transport 1.4. What is achieved in this paper Description of the PCR3BP Global Dynamics 2.1. Problem description 2.2. Equations of motion 2.3. Energy manifolds Computing Transport 3.1. Lobe dynamics 3.1.1. Boundaries, regions, pips, lobes, and turnstiles defined 3.1.2. Multilobe, self-intersecting turnstiles 3.1.3. Expressions for the transport of species 3.2. Set oriented approach 3.2.1. The transfer operator 3.2.2. Discretization of the transfer operator 3.2.3. Approximation of transport rates 3.2.4. Convergence 3.2.5. Almost invariant decompositions 3.2.6. Graph formulation 3.2.7. Heuristics and tools for the graph partitioning problem Example: The Sun-Jupiter-Asteroid System 4.1. Lobe dynamics 4.1.1. Symmetries of the Poincare map / 4.1.2. Finding a fixed point p of / 4.1.3. Finding the stable and unstable manifolds of p under / 4.1.4. Defining the regions and finding the relevant lobes 4.1.5. Higher iterates of the map 4.1.6. Re-entrainment of the lobes 4.2. Set oriented approach 4.2.1. Almost invariant decomposition of the Poincare section 4.2.2. Transport for a two-set partition 4.2.3. Local optimization 4.2.4. Extrapolation 4.2.5. Higher iterates of the map 4.2.6. Return times of the Poincare map Conclusions and Future Directions 5.1. Good agreement between approaches 5.2. Extension to higher dimensions and time dependent systems 5.3. Merging techniques into a single software package 5.4. Miscellany 5.5. Progress towards the grand challenges in computational science

1. Introduction The mathematical description of transport phenomena applies to a wide range of physical systems across many scales [Meiss, 1992; Wiggins, 1992;

6 7 7 8 8 8 8 9 10 10 11 11 12 13 13 14 14 16 17 17 18 19 19 19 19 19 20 20 21 22 22 22 24 25 26 27 27 27 27 28 28 28

Rom-Kedar, 1999]. The recent and surprisingly effective application of methods combining dynamical systems ideas with those from chemistry to the transport of Mars impact ejecta underlines

Transport in Dynamical Astronomy

this point [Jaffe et al, 2002]. In this paper, we develop computational methods to study transport based on the relationship between statistics and geometry in a nonlinear dynamical system with mixed regular and chaotic motion. Our focus is on the transport of material throughout the solar system. However, these methods are fundamental and broad-based; they may be applied to diverse areas of study, including fluid mixing [Rom-Kedar et al, 1990; Malhotra & Wiggins, 1998; Poje & Haller, 1999; Coulliette & Wiggins, 2001; Lekien et al, 2003], iV-body problems in physical chemistry [Jaffe et al, 2000; Lekien &; Marsden, 2004] as well as other problems in dynamical astronomy. For example, the recent discovery of several binary pairs in the asteroid and Kuiper belts has stimulated interest in computing the formation and dissociation rates of such binary pairs (see, e.g. [Goldreich et al, 2002; Scheeres, 2002; Scheeres et al., 2002; Veillet et al, 2002]). Dynamical processes in the solar

system

Our understanding of the solar system has changed dramatically in the past several decades with the realization that the orbits of the planets and some minor bodies are chaotic. In the case of planets, this chaos is of a sufficiently weak nature that their motion appears quite regular on relatively short time scales [Laskar, 1989]. In contrast, small bodies such as asteroids, comets, and Kuiperbelt objects can exhibit strongly chaotic motion through their interactions with the planets and the Sun, exhibiting Lyapunov times of only a few decades [Torbett

(1)

where U =

x2 + y2 2

1- e ~r~s

e rp

e(l - e) 2 '

Here, the subscripts of U denote partial differentiation in the respective variable, and rs,rp are the distances from the particle to the Sun and planet, respectively. See [Szebehely, 1967] for more details on the derivation of this equation and [Koon et al., 2004] for its derivation using Lagrangian mechanics.

2.3.

Energy

manifolds

Equations (1) are autonomous and are in EulerLagrange form (and thus, using the Legendre transformation, can be put into Hamiltonian form as well). They have an energy integral E = -(x2 + y2) + U(x,y),

y = 0,

I \

E<Ei

(b) Case 2: E\ < E < Ei

y>0,

x