Kurths / Zhou Synchronization in Oscillatory Networks

1. Auflage 2007
ISBN: 978-3-540-71269-5
Verlag: Springer-Verlag
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 370 Seiten

This work systematically investigates a large number of oscillatory network configurations that are able to describe many real systems such as electric power grids, lasers or even the heart muscle, to name but a few. The book is conceived as an introduction to the field for graduate students in physics and applied mathematics as well as being a compendium for researchers from any field of application interested in quantitative models.

Autoren/Hrsg.

Kurths, Jürgen

Zhou, Changsong

Inhaltsverzeichnis

1;Preface;7
2;Contents;9
3;Basics on Synchronization and Paradigmatic Models;15
3.1;1 Introduction;16
3.1.1;1.1 Synchronization Phenomena in Nature, Physics, and Engineering;16
3.1.2;1.2 Goal of the Book;18
3.1.3;1.3 Terminological Remarks;20
3.1.4;1.4 Bibliographical Remarks;21
3.2;2 Basic Models;23
3.2.1;2.1 Harmonic Oscillator: Amplitude, Frequency and Phase of Oscillations;23
3.2.2;2.2 Van der Pol Oscillator: Quasi-Harmonic and Relaxation Limit Cycles;24
3.2.3;2.3 Rössler Oscillator: From Phase-Coherent to Funnel Chaotic Attractors;26
3.2.4;2.4 Lorenz Oscillator: Classic and Intermittent Chaotic Attractors;30
3.2.5;2.5 Phase Oscillators;33
3.2.6;2.6 Discrete Map for Spiking–Bursting Neural Activity;40
3.2.7;2.7 Excitable Systems;41
3.3;3 Synchronization Due to External Periodic Forcing;47
3.3.1;3.1 Synchronization of Limit-Cycle Oscillator by External Force;48
3.3.2;3.2 Phase Synchronization of a Chaotic Rössler Oscillator by External Driving;51
3.3.3;3.3 Imperfect Phase Synchronization;54
3.3.4;3.4 Transition to the Regime of Chaotic Phase Synchronization: The Role of Unstable Periodic Orbits;57
3.3.5;3.5 External Phase Synchronization of Chaotic Intermittent Oscillators;59
3.3.6;3.6 Synchronous Response of Excitable Systems to a Periodic External Force;64
3.3.7;3.7 Conclusions;65
3.4;4 Synchronization of Two Coupled Systems;67
3.4.1;4.1 Synchronization of Regular Systems;67
3.4.2;4.2 Synchronization of Coupled Chaotic Oscillators;80
3.4.3;4.3 Synchronization of Coupled Circle Maps;102
4;Synchronization in Geometrically Regular Ensembles;112
4.1;5 Ensembles of Phase Oscillators;113
4.1.1;5.1 General Model and Malkin’s Theorem;114
4.1.2;5.2 Unidirectional Coupling;116
4.1.3;5.3 Synchronization Phenomena in a Chain of Bidirectionally Coupled Phase Oscillators;122
4.1.4;5.4 Influence of Non-Uniform Rotations on the Synchronization;131
4.1.5;5.5 Mutual Entrainment in Populations of Globally Coupled Phase Oscillators;133
4.1.6;5.6 Synchronization Phenomena in a Chain of Coupled Pendulum- Like Equations;135
4.1.7;5.7 Conclusions;137
4.2;6 Chains of Coupled Limit-Cycle Oscillators;139
4.2.1;6.1 Objectives;140
4.2.2;6.2 Synchronization Clusters and Multistability at Linear Variation of Individual Frequencies Along the Chain;140
4.2.3;6.3 Oscillation Death;153
4.2.4;6.4 Effects of Nonuniformity of the Frequency Mismatch Gradient in the Formation of Synchronized Clusters;155
4.2.5;6.5 Synchronization in a Chain of van der Pol Oscillators;157
4.2.6;6.6 Conclusions;160
4.3;7 Ensembles of Chaotic Oscillators with a Periodic- Doubling Route to Chaos, Rössler Oscillators;161
4.3.1;7.1 Synchronization Effects in a Homogeneous Chain of Rössler Oscillators;161
4.3.2;7.2 Basic Model of a Nonhomogeneous Chain, Phase and Frequency Definitions, and Criteria of Phase Synchronization;162
4.3.3;7.3 Phase Synchronization in a Chain with a Linear Distribution of Natural Frequencies, Phase- Coherent Rössler Oscillators;164
4.3.4;7.4 Synchronization in a Chain with Randomly Distributed Natural Frequencies;170
4.3.5;7.5 Phase Synchronization of Rössler Oscillators with the Funnel Attractor;172
4.3.6;7.6 Anomalous Collective Behavior of Coupled Chaotic Oscillators;175
4.3.7;7.7 Conclusions;177
4.4;8 Synchronization of Intermittent-Like Oscillations in Chains of Coupled Maps;179
4.4.1;8.1 Model of Coupled Intermittent Maps, Phase and Frequency, Synchronization Criteria;180
4.4.2;8.2 Linearly Distributed Control Parameters, Soft Transition to Global Synchronization Regime;181
4.4.3;8.3 Randomly Distributed Control Parameter, Transition to Spatiotemporal Intermittency;183
4.4.4;8.4 Collective Oscillations in a Chain of Spiking Maps;187
4.4.5;8.5 Synchronization in Ensembles of Globally Coupled Bursting Oscillators;188
4.4.6;8.6 Conclusions;195
4.5;9 Regular and Chaotic Phase Synchronization of Coupled Circle Maps;197
4.5.1;9.1 Common Model for a Chain of Coupled Circle Maps;198
4.5.2;9.2 Synchronization in a Chain of Identical Circle Maps;199
4.5.3;9.3 Ensembles of Coupled Nonidentical Circle Maps and Criteria of Synchronization;209
4.5.4;9.4 Synchronization and Clustering in a Chain of Regular CMs;210
4.5.5;9.5 Chaotic Phase Synchronization;217
4.5.6;9.6 Conclusions;218
4.6;10 Controlling Phase Synchronization in Oscillatory Networks;222
4.6.1;10.1 General Principles of Automatic Synchronization;223
4.6.2;10.2 Two Coupled Poincare Systems;225
4.6.3;10.3 Coupled van der Pol and Rössler Oscillators;226
4.6.4;10.4 Two Coupled Rössler Oscillators;229
4.6.5;10.5 Coupled Rössler and Lorenz Oscillators;232
4.6.6;10.6 Principles of Automatic Synchronization in Networks of Coupled Oscillators;233
4.6.7;10.7 Synchronization of Locally Coupled Regular Oscillators;234
4.6.8;10.8 Synchronization of Locally Coupled Chaotic Oscillators;237
4.6.9;10.9 Synchronization of Globally Coupled Chaotic Oscillators;239
4.6.10;10.10 Conclusions;240
4.7;11 Chains of Limit-Cycle Oscillators;242
4.7.1;11.1 Introduction and Model;242
4.7.2;11.2 Mechanism of Localized Structure Formation;244
4.7.3;11.3 Dissipative Coupling (Zero Dispersion );244
4.7.4;11.4 Nonscalar (Dissipative and Conservative) Coupling;250
4.7.5;11.5 Conclusions;257
4.8;12 Chains and Lattices of Excitable Luo– Rudy Systems;259
4.8.1;12.1 Objectives;260
4.8.2;12.2 Cardiac Model;261
4.8.3;12.3 Methods: Theoretical Basis;262
4.8.4;12.4 Computational Results;263
4.8.5;12.5 Conclusions;273
5;Synchronization in Complex Networks and Influence of Noise;275
5.1;13 Noise-Induced Synchronization in Ensembles of Oscillatory and Excitable Systems;276
5.1.1;13.1 Degrading Effects of Noise: Noise-Induced Phase Slips;277
5.1.2;13.2 Noise-Induced CS and PS in Uncoupled Chaotic Oscillators;280
5.1.3;13.3 Noise-Enhanced PS in Weakly Coupled Chaotic Oscillators;295
5.1.4;13.4 Noise-Enhanced Synchronization-Like Phenomena in Arrays of Coupled Excitable Cells;312
5.1.5;13.5 Conclusions;322
5.2;14 Networks with Complex Topology;323
5.2.1;14.1 Introduction;323
5.2.2;14.2 Dynamical Equations and Stability Analysis;326
5.2.3;14.3 Phase Synchronization in Small-World Networks of Oscillators;327
5.2.4;14.4 Synchronization in Scale-Free Networks of Oscillators;330
5.2.5;14.5 Mean-Field Analysis of Hierarchical Synchronization;337
5.2.6;14.6 Synchronization Properties of Weighted Networks;338
5.2.7;14.7 Conclusions;345
6;Glossary;347
7;Acknowledgments;348
8;References;349
9;Index;366