Modeling And Computations in Dynamical Systems: In Commemoration Of The 100th Anniversary Of The Birth Of John von Neumann

357 Pages·2006·38.74 MB·English

by  et al Eusebius J. Doedel (Editor)

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
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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 *•* > e«-!«„ D UAI •» Q
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NONLINEAR SCIENCE
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
Concordia university, Canada
Gabor Domokos
Budapest university of Technology and Economics, Hungary
loannis G. Kevrekidis
Princeton university, USA
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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.
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CONTENTS
Editorial 1
Transport in Dynamical Astronomy and Multibody Problems 3
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
A Brief Survey on the Numerical Dynamics for Functional Differential
Equations 33
B. M. Garay
Bifurcations and Continuous Transitions of Attractors in Autonomous and
Nonautonomous Systems 47
P. E. Kloeden and S. Siegmund
A Survey of Methods for Computing (Un)Stable Manifolds of Vector Fields 67
B. Krauskopf, H. M. Osinga, E. J. Doedel, M. E. Henderson, J. Guckenheimer,
A. Vladimir sky, M. Dellnitz and 0. Junge
Commutators of Skew-Symmetric Matrices 97
A. M. Bloch and A. Iserles
Simple Neural Networks that Optimize Decisions 107
E. Brown, J. Gao, P. Holmes, R. Bogacz, M. Gilzenrat and J. D. Cohen
Newton Flow and Interior Point Methods in Linear Programming 131
J.-P. Dedieu and M. Shub
Numerical Continuation of Branch Points of Equilibria and Periodic Orbits 145
E. J. Doedel, W. Govaerts, Yu. A. Kuznetsov and A- Dhooge
Coarse-Grained Observation of Discretized Maps 165
G. Domokos
Multiple Helical Perversions of Finite, Intristically Curved Rods 175
G. Domokos and T. J. Healey
Bifurcations of Stable Sets in Noninvertible Planar Maps 195
J. P. England, B. Krauskopf and H. M. Osinga
Multiparametric Bifurcations m an Enzyme-Catalyzed Reaction Model 209
E. Freire, b. Pizarro, A. J. Rodriguez-buis and F. Fernandez-Sanchez
v
Straightforward Computation of Spatial Equilibria of Geometrically Exact
Cosserat Rods 253
T. J. Healey and P. G. Mehta
Multiparameter Parallel Search Branch Switching 271
M. E. Henderson
Equation-Free, Effective Computation for Discrete Systems: A Time Stepper
Based Approach 279
J. Moiler, 0. Runborg, P. G. Kevrekidis, K. Lust and I. G. Kevrekidis
Model Reduction for Fluids, Using Balanced Proper Orthogonal
Decomposition 301
C. W. Rowley
Bifurcation Tracking Algorithms and Software for Large Scale Applications 319
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 337
G. Swirszcz
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, the
oreticians 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 spe
cial 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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TRANSPORT IN DYNAMICAL ASTRONOMY
AND MULTIBODY PROBLEMS
MICHAEL DELLNITZ*, OLIVER JUNGE*, WANG SANG KOONt,
FRANCOIS LEKIEN*, MARTIN W. LO§, JERROLD E. MARSDEN^,
KATHRIN PADBERG*, ROBERT 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 invari
ant sets, resonance regions as well as transport rates and bottlenecks between regions in dynam
ical 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 4
1.1. Need for modification of current transport calculations 6
1.1.1. Chemistry 6
1.1.2. Dynamical astronomy 6
1.2. Current methods for the study of transport in the PCR3BP 6
1.2.1. Analytical methods: single resonance theory and resonance
overlap criterion 6
3