| Basics on Synchronization and Paradigmatic Models | |
| Introduction | p. 3 |
| Synchronization Phenomena in Nature, Physics, and Engineering | p. 3 |
| Goal of the Book | p. 5 |
| Terminological Remarks | p. 7 |
| Bibliographical Remarks | p. 8 |
| Basic Models | p. 11 |
| Harmonic Oscillator: Amplitude, Frequency and Phase of Oscillations | p. 11 |
| Van der Pol Oscillator: Quasi-Harmonic and Relaxation Limit Cycles | p. 12 |
| Rössler Oscillator: From Phase-Coherent to Funnel Chaotic Attractors | p. 14 |
| Lorenz Oscillator: ôClassicö and Intermittent Chaotic Attractors | p. 18 |
| Phase Oscillators | p. 21 |
| First-Order Phase Oscillator (Active Rotator) | p. 21 |
| Second-Order Phase Oscillator (Pendulum-Like System) | p. 22 |
| Third-Order Phase Oscillator (Chaotic Rotator) | p. 24 |
| Discrete-Time Rotator (Circle Map) | p. 24 |
| Discrete Map for Spiking-Bursting Neural Activity | p. 28 |
| Excitable Systems | p. 29 |
| Hodgkin-Huxley Model | p. 29 |
| FitzHugh-Nagumo Model | p. 30 |
| Luo-Rudy Model | p. 33 |
| Synchronization Due to External Periodic Forcing | p. 35 |
| Synchronization of Limit-Cycle Oscillator by External Force | p. 36 |
| Weak Forcing: Phase Description | p. 36 |
| Synchronization of a van der Pol Oscillator by External Force | p. 37 |
| Phase Synchronization of a Chaotic Rössler Oscillator by External Driving | p. 39 |
| Imperfect Phase Synchronization | p. 42 |
| Transition to the Regime of Chaotic Phase Synchronization: The Role of Unstable Periodic Orbits | p. 45 |
| External Phase Synchronization of Chaotic Intermittent Oscillators | p. 47 |
| Forced Model Quadratic Map | p. 47 |
| Forced Lorenz Oscillator | p. 51 |
| Synchronous Response of Excitable Systems to a Periodic External Force | p. 52 |
| Conclusions | p. 53 |
| Synchronization of Two Coupled Systems | p. 55 |
| Synchronization of Regular Systems | p. 55 |
| Phase Dynamics Approach | p. 56 |
| Synchronization of Two Coupled van der Pol Oscillators | p. 58 |
| Synchronization of Coupled Active Rotators | p. 66 |
| Synchronization of Coupled Chaotic Oscillators | p. 68 |
| Phase Synchronization of Rössler Oscillators | p. 68 |
| Synchronization of Coupled Intermittent Oscillators | p. 77 |
| Oscillatory and Rotatory Synchronization of Chaotic Phase Systems | p. 79 |
| Synchronization of Coupled Circle Maps | p. 90 |
| Regular Synchronization | p. 91 |
| Chaotic Synchronization | p. 93 |
| Synchronization in Geometrically Regular Ensembles | |
| Ensembles of Phase Oscillators | p. 103 |
| General Model and Malkin's Theorem | p. 104 |
| Unidirectional Coupling | p. 106 |
| Synchronization Phenomena in a Chain of Bidirectionally Coupled Phase Oscillators | p. 112 |
| Synchronization, Clustering and Multistability in Chains with Linearly Distributed Individual Frequencies | p. 114 |
| Synchronization Transitions in Chains with Randomly Distributed Individual Frequencies | p. 119 |
| Influence of Non-Uniform Rotations on the Synchronization | p. 121 |
| Mutual Entrainment in Populations of Globally Coupled Phase Oscillators | p. 123 |
| Synchronization Phenomena in a Chain of Coupled Pendulum-Like Equations | p. 125 |
| Conclusions | p. 127 |
| Chains of Coupled Limit-Cycle Oscillators | p. 129 |
| Objectives | p. 130 |
| Synchronization Clusters and Multistability at Linear Variation of Individual Frequencies Along the Chain | p. 130 |
| Model Equations | p. 131 |
| Global Synchronization in an Assembly, Stationary Phase Distributions, Synchronization area | p. 133 |
| Regimes of Cluster Synchronization | p. 135 |
| Multistability | p. 141 |
| Oscillation Death | p. 143 |
| Effects of Nonuniformity of the Frequency Mismatch Gradient in the Formation of Synchronized Clusters | p. 145 |
| Sensitivity of the Structures to Regular Nonuniformities | p. 145 |
| The Effect of Random Dispersion of Individual Frequencies on Cluster Synchronization | p. 146 |
| Synchronization in a Chain of van der Pol Oscillators | p. 147 |
| Conclusions | p. 150 |
| Ensembles of Chaotic Oscillators with a Periodic-Doubling Route to Chaos, Rössler Oscillators | p. 151 |
| Synchronization Effects in a Homogeneous Chain of Rössler Oscillators | p. 151 |
| Basic Model of a Nonhomogeneous Chain, Phase and Frequency Definitions, and Criteria of Phase Synchronization | p. 152 |
| Phase Synchronization in a Chain with a Linear Distribution of Natural Frequencies, Phase-Coherent Rössler Oscillators | p. 154 |
| Theoretical Study | p. 154 |
| Numerical Results | p. 155 |
| Synchronization in a Chain with Randomly Distributed Natural Frequencies | p. 160 |
| Phase Synchronization of Rössler Oscillators with the Funnel Attractor | p. 162 |
| Anomalous Collective Behavior of Coupled Chaotic Oscillators | p. 165 |
| Conclusions | p. 167 |
| Intermittent-Like Oscillations in Chains of Coupled Maps | p. 169 |
| Model of Coupled Intermittent Maps, Phase and Frequency, Synchronization Criteria | p. 170 |
| Linearly Distributed Control Parameters, Soft Transition to Global Synchronization Regime | p. 171 |
| Randomly Distributed Control Parameter, Transition to Spatiotemporal Intermittency | p. 173 |
| Collective Oscillations in a Chain of Spiking Maps | p. 177 |
| Synchronization in Ensembles of Globally Coupled Bursting Oscillators | p. 178 |
| Mutual Synchronization | p. 180 |
| External Synchronization | p. 182 |
| Conclusions | p. 185 |
| Regular and Chaotic Phase Synchronization of Coupled Circle Maps | p. 187 |
| Common Model for a Chain of Coupled Circle Maps | p. 188 |
| Synchronization in a Chain of Identical Circle Maps | p. 189 |
| Symmetrically Coupled Maps | p. 190 |
| Effect of Asymmetry of Coupling | p. 195 |
| Synchronization in Lattices of Coupled Maps | p. 197 |
| Ensembles of Coupled Nonidentical Circle Maps and Criteria of Synchronization | p. 199 |
| Synchronization and Clustering in a Chain of Regular CMs | p. 200 |
| Linear Distribution of Individual Frequencies | p. 200 |
| Random Distribution of Individual Frequencies | p. 206 |
| Chaotic Phase Synchronization | p. 207 |
| Conclusions | p. 208 |
| Controlling Phase Synchronization in Oscillatory Networks | p. 213 |
| General Principles of Automatic Synchronization | p. 214 |
| Two Coupled Poincaré Systems | p. 216 |
| Coupled van der Pol and Rössler Oscillators | p. 217 |
| Two Coupled Rössler Oscillators | p. 220 |
| Coupled Rössler and Lorenz Oscillators | p. 223 |
| Principles of Automatic Synchronization in Networks of Coupled Oscillators | p. 224 |
| Synchronization of Locally Coupled Regular Oscillators | p. 225 |
| Synchronization of Locally Coupled Chaotic Oscillators | p. 228 |
| Synchronization of Globally Coupled Chaotic Oscillators | p. 230 |
| Conclusions | p. 231 |
| Chains of Limit-Cycle Oscillators | p. 233 |
| Introduction and Model | p. 233 |
| Mechanism of Localized Structure Formation | p. 235 |
| Dissipative Coupling (Zero ôDispersionö) | p. 235 |
| Desynchronization of Front Propagation | p. 235 |
| Localized Synchronization Structures | p. 237 |
| Nonlocal Synchronization in Nonhomogeneous Chains | p. 238 |
| Fully Incoherent (Turbulent-Like) Oscillations | p. 239 |
| Nonscalar (Dissipative and Conservative) Coupling | p. 241 |
| Bursting Structures | p. 241 |
| Nonpropagation to Propagation Transition via Intermittency | p. 242 |
| Noise Influence | p. 247 |
| Conclusions | p. 248 |
| Chains and Lattices of Excitable Luo-Rudy Systems | p. 251 |
| Objectives | p. 252 |
| Cardiac Model | p. 253 |
| Methods: Theoretical Basis | p. 254 |
| Computational Results | p. 255 |
| One-Dimensional Simulations | p. 255 |
| Two-Dimensional Simulations | p. 261 |
| Conclusions | p. 265 |
| Synchronization in Complex Networks and Influence of Noise | |
| Noise-Induced Synchronization in Ensembles of Oscillatory and Excitable Systems | p. 269 |
| Degrading Effects of Noise: Noise-Induced Phase Slips | p. 270 |
| Noise-Induced CS and PS in Uncoupled Chaotic Oscillators | p. 273 |
| Noise-Induced CS of Identical Chaotic Oscillators | p. 273 |
| Noise-Induced PS of Nonidentical Uncoupled Chaotic Systems | p. 285 |
| Noise-Enhanced PS in Weakly Coupled Chaotic Oscillators | p. 288 |
| Noise-Enhancecl PS of a Chaotic Laser Due to Periodic Forcing | p. 289 |
| Noise-Enhanced PS of Two Coupled Rössler Oscillators | p. 292 |
| Noise-Enhanced PS in Arrays of Globally Coupled Rössler Oscillators | p. 295 |
| Experimental Observation of Noise-Enhanced PS | p. 297 |
| Noise-Enhanced Synchronization-Like Phenomena in Arrays of Coupled Excitable Cells | p. 305 |
| Phase Synchrony in Chains of Coupled Noisy Excitable Neurons | p. 305 |
| Noise-Enhanced PS of Coupled Excitable Neurons by External Forcing | p. 305 |
| Resonant Pattern Formation in 2D Arrays | p. 313 |
| Conclusions | p. 31 |
| Networks with Complex Topology | p. 317 |
| Introduction | p. 317 |
| Dynamical Equations and Stability Analysis | p. 320 |
| Phase Synchronization in Small-World Networks of Oscillators | p. 321 |
| Synchronization in Scale-Free Networks of Oscillators | p. 324 |
| Mean-Field Analysis of Hierarchical Synchronization | p. 331 |
| Synchronization Properties of Weighted Networks | p. 332 |
| Conclusions | p. 335 |
| Glossary | p. 341 |
| Acknowledgments | p. 343 |
| References | p. 345 |
| Index | p. 363 |
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