MODELING BY NONLINEAR DIFFERENTIAL EQUATIONS

Transcription

1 MODELING BY NONLINEAR DIFFERENTIAL EQUATIONS Dissipative and Conservative Processes

2 WORLD SCIENTIFIC SERIES ON NONLINEAR SCIENCE Editor: Leon O. Chua University of California, Berkeley Series A. Volume 51: Volume 52: Volume 53: Volume 54: Volume 55: Volume 56: Volume 57: Volume 58: Volume 59: Volume 60: Volume 61: Volume 62: Volume 63: Volume 64: Volume 65: Volume 66: Volume 67: Volume 68: MONOGRAPHS AND TREATISES* Symmetry and Complexity K. Mainzer Applied Nonlinear Time Series Analysis M. Small Bifurcation Theory and Applications T. Ma & S. Wang Dynamics of Crowd-Minds A. Adamatzky Control of Homoclinic Chaos by Weak Periodic Perturbations R. Chacón Strange Nonchaotic Attractors U. Feudel, S. Kuznetsov & A. Pikovsky A Nonlinear Dynamics Perspective of Wolfram's New Kind of Science L. O. Chua New Methods for Chaotic Dynamics N. A. Magnitskii & S. V. Sidorov Equations of Phase-Locked Loops J. Kudrewicz & S. Wasowicz Smooth and Nonsmooth High Dimensional Chaos and the Melnikov-Type Methods J. Awrejcewicz & M. M. Holicke A Gallery of Chua Attractors (with CD-ROM) E. Bilotta & P. Pantano Numerical Simulation of Waves and Fronts in Inhomogeneous Solids A. Berezovski, J. Engelbrecht & G. A. Maugin Advanced Topics on Cellular Self-Organizing Nets and Chaotic Nonlinear Dynamics to Model and Control Complex Systems R. Caponetto, L. Fortuna & M. Frasca Control of Chaos in Nonlinear Circuits and Systems B. W.-K. Ling, H. H.-C. Lu & H. K. Lam Chua s Circuit Implementations: Yesterday, Today and Tomorrow L. Fortuna, M. Frasca & M. G. Xibilia Differential Geometry Applied to Dynamical Systems J.-M. Ginoux Determining Thresholds of Complete Synchronization, and Application A. Stefanski A Nonlinear Dynamics Perspective of Wolfram New Kind of Science (Volume III) L. O. Chua * To view the complete list of the published volumes in the series, please visit:

3 WORLD SCIENTIFIC SERIES ON NONLINEAR SCIENCE Series Editor: Leon O. Chua Series A Vol. 69 MODELING BY NONLINEAR DIFFERENTIAL EQUATIONS Dissipative and Conservative Processes Paul E. Phillipson University of Colorado, USA Peter Schuster Universität Wien, Austria World Scientific NEW JERSEY LONDON SINGAPORE BEIJING SHANGHAI HONG KONG TAIPEI CHENNAI

4 Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore USA office: 27 Warren Street, Suite , Hackensack, NJ 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. 69 MODELING BY NONLINEAR DIFFERENTIAL EQUATIONS Dissipative and Conservative Processes 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. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher. ISBN ISBN Printed in Singapore.

5 To our wives, Patricia and Inge

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7 Contents Acknowledgments 1. Theme and Contents of this Book 1 2. Processes in Closed and Open Systems Introduction Thermodynamics of general systems Chemical reactions Autocatalysis in closed and open systems Autocatalysis in closed systems Autocatalysis in the flow reactor Dynamics of Molecular Evolution Introduction Selection and evolution Template induced autocatalysis Autocatalytic oligomerization Biopolymer replication Replication and selection Replication and mutation Error thresholds Replicator equations Schlögl model Fisher s selection equation Symbioses and hypercycles Unlimited growth and selection xi vii

8 viii Modeling by Nonlinear Differential Equations 4. Relaxation Oscillations Introduction Self-exciting relaxation oscillations van der Pol equation Stoker-Haag equation Current induced neuron oscillations Bistability and complex structure of harmonically forced relaxation oscillations Order and Chaos Introduction One dimensional maps Formation of a period window Stability of a period window Topology of one dimensional maps Lorenz equations Low dimensional autocatalytic networks Chua equations Reaction Diffusion Dynamics Introduction Pulse front solutions of Fisher and related equations Diffusion driven spatial inhomogeneities Turing mechanism of chemical pattern formation Solitons Introduction One dimensional lattice dynamics Korteweg-de Vries equation sine-gordon equation Burgers equation Neuron Pulse Propagation Introduction Properties of a neural pulse FitzHugh-Nagumo equations

9 Contents ix 8.4 Hodgkin-Huxley equations An overview Time Reversal, Dissipation and Conservation Introduction Irreversibility and diffusion Theory of random walk Langevin equation and equilibrium fluctuations Newtonian mechanics and asymptotic irreversibility Reversibility and time recurrence A linear synchronous system Recurrence in nonlinear Hamiltonian systems: Fermi-Pasta-Ulam Model Complex dynamics and chaos in Newtonian dynamics: Hénon-Heiles equations Bibliography 213 Index 223

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11 Acknowledgments One of the authors (P.E.P.) wishes to express his appreciation to the Austrian Academy of Sciences for financial support of this work and to the University of Vienna for their hospitality where portions of this book were written. xi