Synchronization in Oscillatory Networks (eBook)

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