## Synchronization: A Universal Concept in Nonlinear SciencesSystems as diverse as clocks, singing crickets, cardiac pacemakers, firing neurons and applauding audiences exhibit a tendency to operate in synchrony. These phenomena are universal and can be understood within a common framework based on modern nonlinear dynamics. The first half of this book describes synchronization without formulae, and is based on qualitative intuitive ideas. The main effects are illustrated with experimental examples and figures, and the historical development is also outlined. The second half of the book presents the main effects of synchronization in a rigorous and systematic manner, describing both classical results on synchronization of periodic oscillators, and recent developments in chaotic systems, large ensembles, and oscillatory media. |

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### Contents

Chapter 1 Introduction | 1 |

Part I Synchronization without formulae | 25 |

Part II Phase locking and frequency entrainment | 173 |

Part III Synchronization of chaotic systems | 299 |

Appendices | 355 |

371 | |

405 | |

### Common terms and phrases

attractor autonomous bifurcation cells chaos chaotic oscillators chaotic systems circle map clocks complete synchronization consider constant corresponds coupled oscillators curve depends described diffusion discuss distribution driven system effect ensemble entrainment equation example external force Figure fixed point fluctuations function globally coupled Hilbert transform illustrate interaction interval isochronous Josephson junctions laser lattice limit cycle linear Lorenz model Lorenz system Lyapunov exponent mean field mean frequency modulation motion natural frequencies noise observed frequency obtain oscillatory pacemaker parameters particle pendulum periodic force periodic orbits periodic oscillations perturbation phase difference phase dynamics phase locking phase plane phase point phase shift phase slips phase space phase synchronization Pikovsky Poincar´e map properties pulse quasiperiodic regime relaxation oscillators rhythm rotation number saddle-node bifurcation Section self-sustained oscillators shown in Fig signal solution stable stroboscopic symmetric synchronization region synchronization transition threshold trajectories unstable variable zero