## Chaos in Systems with NoiseAs in the first edition, the influence of random noise on the chaotic behavior of dissipative dynamical systems is investigated. Problems are illustrated by mechanical examples. This revised and updated edition contains new sections on the summary of probability theory, homoclinic chaos, Melnikov method, routes to chaos, stabilization of period-doubling, and Hopf bifurcation by noise. Some chapters have been rewritten and new examples have been added. |

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### Contents

Preface to the second edition vl11 | 1 |

Chaotic and Stochastic Processes | 19 |

Noisy Dynamical Systems | 25 |

FokkerPlanckKolmogorov Equation | 33 |

MultiMaxima Probability Density Functions | 42 |

Random Lyapunov Exponents | 53 |

Poincare Maps for Noisy Systems | 65 |

Regular and Chaotic Stochastic Processes | 76 |

Stochastic Sensitivity Functions and Chaos | 95 |

Examples | 117 |

Noisy Routes to Chaos | 154 |

References | 219 |

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### Common terms and phrases

amplitude approximation Arneodo asymptotic behaviour attractor averaged band-limited white noise bifurcation diagram chaotic stochastic process constant Coullet Definition described deterministic system differential equation dynamical system eigenvalue example exists frequency homoclinic orbit Hopf bifurcation hyperbolic fixed point initial conditions initial value problem interval Kapitaniak kind region Kolmogorov entropy let us consider Lett linear logistic map loss of chaos Math maximum Lyapunov exponent mean Poincare maps mean value Melnikov function Naschie neighbourhood noise intensity noisy system nonlinear obtain oscillator parameter period-doubling bifurcation periodic orbit perturbed phase space Phys Physica Power spectra probability density function process x(t,u Prog random noise random variable realizations regular stochastic process route to chaos shown in Figure shows chaotic behaviour Sound Vibr spatial plot spectral density stable and unstable Steeb stochastic sensitivity function sure solution t+At tangent theorem theory Tresser unperturbed system unstable manifolds unstable region value of noise variational equation zero