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Chaotic Synchronization: Applications to Living Systems (World Scientific Series on Nonlinear Science, 42) PDF

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WORLD SCIENTIFIC ONLINEAR SCIENC Series Editor: Leon O. Chua HiET HPPLICflTIONS TO LIVING SYSTEMS Erik Mosekilde, Yuri Maistrenko & Dmitry Postnov World Scientific CHAOTIC SVNCHRONIZRTIOI APPLICATIONS TO LIVING SYSTEHS WORLD SCIENTIFIC SERIES ON NONLINEAR SCIENCE Editor: Leon O. Chua University of California, Berkeley Series A. MONOGRAPHS AND TREATISES Volume 25: Chaotic Dynamics in Hamiltonian Systems H. Dankowicz Volume 26: Visions of Nonlinear Science in the 21 st Century Edited by J. L Huertas, W.-K. Chen & Ft. N. Madan Volume 27: The Thermomechanics of Nonlinear Irreversible Behaviors — An Introduction G. A. Maugin Volume 28: Applied Nonlinear Dynamics & Chaos of Mechanical Systems with Discontinuities Edited by M. Wiercigroch & B. de Kraker Volume 29: Nonlinear & Parametric Phenomena* V. Damgov Volume 30: Quasi-Conservative Systems: Cycles, Resonances and Chaos A. D. Morozov Volume 31: CNN: A Paradigm for Complexity L. O. Chua Volume 32: From Order to Chaos II L P. Kadanoff Volume 33: Lectures in Synergetics V. I. Sugakov Volume 34: Introduction to Nonlinear Dynamics* L Kocarev & M. P. Kennedy Volume 35: Introduction to Control of Oscillations and Chaos A. L Fradkov & A. Yu. Pogromsky Volume 36: Chaotic Mechanics in Systems with Impacts & Friction B. Blazejczyk-Okolewska, K. Czolczynski, T. Kapitaniak & J. Wojewoda Volume 37: Invariant Sets for Windows — Resonance Structures, Attractors, Fractals and Patterns A. D. Morozov, T. N. Dragunov, S. A. Boykova & O. V. Malysheva Volume 38: Nonlinear Noninteger Order Circuits & Systems — An Introduction P. Arena, R. Caponetto, L Fortuna & D. Porto Volume 39: The Chaos Avant-Garde: Memories of the Early Days of Chaos Theory Edited by Ralph Abraham & Yoshisuke Ueda Volume 40: Advanced Topics in Nonlinear Control Systems Edited by T. P. Leung & H. S. Qin Volume 41: Synchronization in Coupled Chaotic Circuits and Systems C. W. Wu *Forthcoming NONLINEAR SCIENCE •* S^WSA vol.** Series Editor; Leon O. Chua Erik Mosekilde The Technical University of Denmark Yuri Maistrenko National Academy of Sciences, Ukraine Dmitry Postnov Saratov state University, Russia World Scientific *aw Jersey "London'Singapore* Hong Kong Published by World Scientific Publishing Co. Pte. Ltd. P O Box 128, Farrer Road, Singapore 912805 USA office: Suite IB, 1060 Main Street, River Edge, NJ 07661 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. CHAOTIC SYNCHRONIZATION Applications to Living Systems Copyright © 2002 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 981-02-4789-3 Printed in Singapore by Uto-Print PREFACE The cooperative behavior of coupled nonlinear oscillators is of interest in connec- tion with a wide variety of different phenomena in physics, engineering, biology, and economics. Networks of coupled nonlinear oscillators have served as models of spatio-temporal pattern formation and simple forms of turbulence. Systems of coupled nonlinear oscillators may be used to explain how different sectors of the economy adjust their individual commodity cycles relative to one another through the exchange of goods and capital units or via aggregate signals in the form of varying interest rates or raw materials prices. Similarly, in the biolog- ical sciences it is important to understand how a group of cells or functional units, each displaying complicated nonlinear dynamic phenomena, can interact with each other to produce a coordinated response on a higher organizational level. It is well-known, for instance, that waves of synchronized behavior that propagate across the surface of the heart are essential for the muscle cells to act in unison and produce a regular contraction. Waves of synchronized behavior can also be observed to propagate across the insulin producing beta-cells of the pancreas. In many cases the individual oscillators display chaotic dynamics. It has long been recognized, for instance, that the ability of the kidneys to compen- sate for variations in the arterial blood pressure partly rests with controls as- sociated with the individual functional unit (the nephron). The main control is the so-called tubuloglomerular feedback that regulates the incoming blood flow in response to variations in the ionic composition of the fluid leaving the nephron. For rats with normal blood pressure, the individual nephron typically vi Preface exhibits regular limit cycle oscillations in the incoming blood flow. For such rats, both in-phase and antiphase synchronization can be observed between ad- jacent nephrons. For spontaneously hypertensive rats, where the pressure vari- ations for the individual nephron are highly irregular, signs of chaotic phase synchronization are observed. In the early 1980's, Fujisaka and Yamada showed how two identical chaotic oscillators under variation of the coupling strength can attain a state of com- plete synchronization in which the motion of the coupled system takes place on an invariant subspace of total phase space. This type of chaotic synchroniza- tion has subsequently been studied by a significant number of investigators, and a variety of applications for chaos suppression, for monitoring and con- trol of dynamical systems, and for different communication purposes have been suggested. Important questions that arise in this connection concern the stability of the synchronized state to noise or to a small parameter mismatch between the interacting oscillators. Other questions relate to the form of the basin of attraction for the synchronized chaotic state and to the bifurcations through which this state loses its stability. Recent studies of these problems have led to the discovery of a large number of new phenomena, including riddled basins of attraction, attractor bubbling, blowout bifurcations, and on-off intermittency. In addition to various electronic systems, synchronization of interacting chaotic oscillators has been observed for laser systems, for coupled supercon- ducting Josephson junctions, and for interacting electrochemical reactors. For systems of three or more coupled oscillators, one can observe the phenomenon of partial synchronization where some of the oscillators synchronize while others do not. This phenomenon is of interest in connection with the development of new types of communication systems where one mixes a message with a chaotic signal. Primarily through the works of Rosenblum and Pikovsky it has become clear that even systems that are quite different in nature (or oscillators that have dif- ferent parameter settings) can exhibit a form of chaotic synchronization where the phases of the interacting oscillators are locked to move in synchrony whereas the amplitudes can develop quite differently. This phenomenon, referred to as chaotic phase synchronization, is of particular importance for living systems where the interacting functional units cannot be assumed to be identical. Kuramoto and Kaneko have initiated the study of clustering in large en- Preface vii sembles of interacting chaotic oscillators with a so-called global (i.e., all-to-all) coupling structure. This type of analyses is relevant for instance to economic sectors that interact via the above mentioned aggregate variations in interest rates and raw materials prices. However, biological systems also display many examples of globally coupled oscillators. The beta-cells in the pancreas, for instance, respond to variations in the blood glucose concentration, variations that at least partly are brought about by changes in the cells' aggregate release of insulin. Important questions that arise in this connection relate to the way in which the clusters are formed and break up as the coupling between the oscillators is varied. The purpose of the book is to present and analyze some of the many interest- ing new phenomena that arise in connection with the interaction of two or more chaotic oscillators. Among the subjects that we treat are periodic orbit thresh- old theory, weak stability of chaotic states, and the formation of riddled basins of attraction. In this connection we discuss local and global riddling, the roles of the absorbing and mixed absorbing areas, attractor bubbling, on-off intermit- tency, and the influence of a small parameter mismatch or of an asymmetry in the coupling structure. We also consider partial synchronization, transitions to chaotic phase synchronization, the role of multistability, coherence resonance, and clustering in ensembles of many noise induced oscillators. However, our aim is also to illustrate how all of these concepts can be ap- plied to improve our understanding of systems of interacting biological oscilla- tors. In-phase synchronization, for instance, where the nephrons of the kidney simultaneously perform the same regulatory adjustments of the incoming blood flow, is expected to produce fast and strong overall reactions to a change in the external conditions. In the absence of synchronization, on the other hand, the response of the system in the aggregate is likely to be slower and less pro- nounced. Hence, part of the regulation of the kidney may be associated with transitions between different states of synchronization among the functional units. Besides synchronization of interacting nephrons, the book also discusses chaotic synchronization and riddled basins of attractions for coupled pancre- atic cells, homoclinic transitions to chaotic phase synchronization in coupled microbiological reactors, and clustering in systems of noise excited nerve cells. To a large extent the book is based on contributions that have been made over the last few years by the Chaos Group at the Technical University of Den- viii Preface mark, by the Department of Mathematics, the National Academy of Sciences of Ukraine in Kiev, and by the Department of Physics, Saratov State University. We would like to thank our collaborators and students Brian Lading, Alexander Balanov, Tanya Vadivasova, Natasha Janson, Alexey Pavlov, Jacob Laugesen, Alexey Taborov, Vladimir Astakhov, Morten Dines Andersen, Niclas Carlsson, Christian Haxholdt, Christian Kampmann, and Carsten Knudsen for the many contributions they have made to the present work. Arkady Pikovsky, Jiirgen Kurths, Michael Rosenblum, Vladimir Belykh, Igor Belykh, Sergey Kuznetsov, Vadim Anishchenko, Morten Colding-J0rgensen, Jeppe Sturis, John D. Ster- man, Laura Gardini, and Christian Mira are acknowledged for many helpful suggestions. We would also like to thank Niels-Henrik Holstein-Rathlau and Kay-Pong Yip who have made their experimented data on coupled nephrons available to us. Most of all, however, we would like to thank Vladimir Maistrenko, Oleksandr Popovych, Sergiy Yanchuk, and Olga Sosnovtseva who have been our closest collaborators in the study of chaotic synchronization. Without the enthusiastic help from these friend and colleagues, the book would never have been possible. The book has appeared at a time when research in chaotic synchronization is virtually exploding, and new concepts and ideas emerge from week to week. Hence, it is clear that we have not been able to cover all the relevant aspects of the field. We hope that the combination of mathematical theory, model formulation, computer simulations, and experimental results can inspire other researchers in this fascinating area. We have tried to make the book readable to students and young scientists without the highest expertise in chaos theory. On the other hand, the reader is assumed to have a good knowledge about the basic concepts and methods of nonlinear dynamics from previous studies. The book is dedicated to Lis Mosekilde. In her short scientific career she became the internationally most respected Danish expert in the fields of bone remodelling and osteoporosis. Lyngby, November 2001 Erik Mosekilde, Yuri Maistrenko and Dmitry Postnov Contents PREFACE v 1 COUPLED NONLINEAR OSCILATORS 1 1.1 The Role of Synchronization 1 1.2 Synchronization Measures 7 1.3 Mode-Locking of Endogenous Economic Cycles 13 2 T R A N S V E R S E STABILITY OF COUPLED M A P S 3 2.1 Ridling, Bubling, and On-Of Intermitency 3 2.2 Weak Stability of the Synchronized Chaotic State 37 2.3 Formation of Ridled Basins of Atraction 41 2.4 Destabilization of Low-Periodic Orbits 4 2.5 Diferent Ridling Scenarios 49 2.6 Intermingled Basins of Atraction 54 2.7 Partial Synchronization for Thre Coupled Maps 56 3 UNFOLDING THE RIDLING BIFURCATION 75 3.1 Localy and Globaly Ridled Basins of Atraction 75 3.2 Conditions for Soft and Hard Ridling 80 3.3 Example of a Soft Ridling Bifurcation 8 3.4 Example of a Hard Ridling Bifurcation 93 3.5 Destabilization Scenario for a — a,\ 95 3.6 Coupled Intermitency-I Maps 104 3.7 The Contact Bifurcation 109 3.8 Conclusions 16 4 TIME-CONTINUOUS SYSTEMS 123 4.1 Two Coupled Rosler Oscilators 123 4.2 Transverse Destabilization of Low-Periodic Orbits 125 ix

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