Chaotic Synchronization

Applications To Living Systems

By: Yuri L Maistrenko, Erik Mosekilde, Dmitry Postnov

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Hardcover | 8 March 2002

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Interacting chaotic oscillators are of interest in many areas of physics, biology, and engineering. In the biological sciences, for instance, one of the challenging problems is to understand how a group of cells or functional units, each displaying complicated nonlinear dynamic phenomena, can interact with one another to produce a coherent response on a higher organizational level. This book is a guide to the concept of chaotic synchronization. The topics covered range from transverse stability and riddled basins of attraction in a system of two coupled logistic maps, over partial synchronization and clustering in systems of many chaotic oscillators, to noise-induced synchronization of coherence resonance oscillators. Other topics treated in the book are on-off intermittency and the role of the absorbing and mixed absorbing areas, periodic orbit threshold theory, the influence of a small parameter mismatch, and different mechanisms for chaotic phase synchronization. The biological examples include synchronization of the bursting behaviour of coupled insulin-producing beta cells, chaotic phase synchronization in the pressure and flow regulation of neighbouring functional units of the kidney, and homoclinic transitions to phase synchronization in microbiological reactors. This book should be of interest to students and researchers interested in applying new concepts of chaotic synchronization and clustering to biological systems.

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Preface	p. v
Coupled Nonlinear Oscillators	p. 1
The Role of Synchronization	p. 1
Synchronization Measures	p. 7
Mode-Locking of Endogenous Economic Cycles	p. 13
Transverse Stability of Coupled Maps	p. 33
Riddling, Bubbling, and On-Off Intermittency	p. 33
Weak Stability of the Synchronized Chaotic State	p. 37
Formation of Riddled Basins of Attraction	p. 41
Destabilization of Low-Periodic Orbits	p. 44
Different Riddling Scenarios	p. 49
Intermingled Basins of Attraction	p. 54
Partial Synchronization for Three Coupled Maps	p. 56
Unfolding the Riddling Bifurcation	p. 75
Locally and Globally Riddled Basins of Attraction	p. 75
Conditions for Soft and Hard Riddling	p. 80
Example of a Soft Riddling Bifurcation	p. 88
Example of a Hard Riddling Bifurcation	p. 93
Destabilization Scenario for a = a[subscript 1]	p. 95
Coupled Intermittency-III Maps	p. 104
The Contact Bifurcation	p. 109
Conclusions	p. 116
Time-Continuous Systems	p. 123
Two Coupled Rossler Oscillators	p. 123
Transverse Destabilization of Low-Periodic Orbits	p. 125
Riddled Basins	p. 130
Bifurcation Scenarios for Asynchronous Cycles	p. 134
The Role of a Small Parameter Mismatch	p. 140
Influence of Asymmetries in the Coupled System	p. 145
Transverse Stability of the Equilibrium Point	p. 147
Partial Synchronization of Coupled Oscillators	p. 154
Clustering in a System of Four Coupled Oscillators	p. 162
Arrays of Coupled Rossler Oscillators	p. 166
Coupled Pancreatic Cells	p. 177
The Insulin Producing Beta-Cells	p. 177
The Bursting Cell Model	p. 181
Bifurcation Diagrams for the Cell Model	p. 185
Coupled Chaotically Spiking Cells	p. 192
Locally Riddled Basins of Attraction	p. 196
Globally Riddled Basins of Attraction	p. 200
Effects of Cell Inhomogeneities	p. 203
Chaotic Phase Synchronization	p. 211
Signatures of Phase Synchronization	p. 211
Bifurcational Analysis	p. 217
Role of Multistability	p. 222
Mapping Approach to Multistability	p. 227
Suppression of the Natural Dynamics	p. 233
Chaotic Hierarchy in High Dimensions	p. 239
A Route to High-Order Chaos	p. 249
Population Dynamic Systems	p. 259
A System of Cascaded Microbiological Reactors	p. 259
The Microbiological Oscillator	p. 262
Nonautonomous Single-Pool System	p. 265
Cascaded Two-Pool System	p. 270
Homoclinic Synchronization Mechanism	p. 274
One-Dimensional Array of Population Pools	p. 280
Conclusions	p. 284
Clustering of Globally Coupled Maps	p. 291
Ensembles of Coupled Chaotic Oscillators	p. 291
The Transcritical Riddling Bifurcation	p. 296
Global Dynamics after a Transcritical Riddling	p. 302
Riddling and Blowout Scenarios	p. 307
Influence of a Parameter Mismatch	p. 313
Stability of K-Cluster States	p. 318
Desynchronization of the Coherent Chaotic State	p. 321
Formation of Nearly Symmetric Clusters	p. 326
Transverse Stability of Chaotic Clusters	p. 329
Strongly Asymmetric Two-Cluster Dynamics	p. 334
Interacting Nephrons	p. 349
Kidney Pressure and Flow Regulation	p. 349
Single-Nephron Model	p. 354
Bifurcation Structure of the Single-Nephron Model	p. 359
Coupled Nephrons	p. 365
Experimental Results	p. 370
Phase Multistability	p. 375
Transition to Synchronous Chaotic Behavior	p. 382
Coherence Resonance Oscillators	p. 395
But What about the Noise?	p. 395
Coherence Resonance	p. 400
Mutual Synchronization	p. 404
Forced Synchronization	p. 408
Clustering of Noise-Induced Oscillations	p. 412
Index	p. 425
Table of Contents provided by Ingram. All Rights Reserved.

Chaotic Synchronization

Applications To Living Systems

At a Glance

Hardcover

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