## Dynamical Chaos: Models and Experiments : Appearance Routes and Structure of Chaos in Simple Dynamical SystemsIn this book, bifurcational mechanisms of the development, structure and properties of chaotic attractors are investigated by numerical and physical experiments based on the methods of the modern theory of nonlinear oscillations. The typical bifurcations of regular and chaotic attractors which are due to parameter variations are analyzed.Regularities of the transition to chaos via the collapse of quasiperiodic oscillations with two and three frequencies are investigated in detail. The book deals with the problems of chaotic synchronization, interaction of attractors and the phenomenon of stochastic resonance. The problems of fluctuation influence on the bifurcations and properties of chaotic attractors are investigated more closely.All principal problems are investigated by the comparison of theoretical and numerical results and data from physical experiments. |

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### Contents

Contents | 1 |

Numerical Methods | 33 |

Inertial Nonlinearity Oscillator | 69 |

Autonomous Oscillation Regimes | 90 |

Quasiattractor Structure | 122 |

TwoFrequency Oscillation | 146 |

Breakdown of Two | 174 |

Synchronization of Chaos | 199 |

Nonlinear Phenomena | 219 |

Bifurcations of Dynamical System | 268 |

Chaos Structure and Properties | 302 |

colored noise | 308 |

Reconstruction of Dynamical | 337 |

### Common terms and phrases

algorithm amplitude approximation basic frequencies bifurcation diagram bifurcation line bifurcation point birth calculation chaos-chaos chaotic attractor characteristic Chua's circuit computer simulation control parameters correlation corresponding critical point defined dependence dimension dynamical system eigenvalues equations equilibrium ergodic evolution experimental finite fixed point full-scale experiments function Henon map homoclinic trajectories inertial nonlinearity intermittency intersection invariant curve investigation laminar phase LCE spectrum limit cycle linear Lorenz Lorenz attractor Lorenz model Lyapunov exponent merging multipliers noise intensity obtained one-dimensional oscillation regimes oscillator with inertial parameter plane parameter values period doubling bifurcations period-one period-two periodic solution perturbation Phase portraits phase space phase trajectory phenomena phenomenon plane of parameters Poincare power spectrum probability density qualitatively quasiattractor realized reconstruction resonant cycle saddle cycles saddle-focus saddle-node bifurcation secant sector separatrix separatrix loop spectra stationary stochastic strange attractor three-dimensional torus breakdown torus-chaos transition to chaos two-dimensional torus unstable manifolds vicinity winding number zero