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

441 Pages·2002·25.05 MB·English

by  Erik Mosekilde, Dmitry Postnov, Yuri Maistrenko, Iu. L. Maistrenko

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
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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*
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Volume 30: Quasi-Conservative Systems: Cycles, Resonances and Chaos
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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
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Volume 36: Chaotic Mechanics in Systems with Impacts & Friction
B. Blazejczyk-Okolewska, K. Czolczynski, T. Kapitaniak & J. Wojewoda
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Edited by Ralph Abraham & Yoshisuke Ueda
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C. W. Wu
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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
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CHAOTIC SYNCHRONIZATION
Applications to Living Systems
Copyright © 2002 by World Scientific Publishing Co. Pte. Ltd.
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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