## Chaos in Systems with NoiseThe influence of random noise on the chaotic behaviour of nonlinear system is investigated. Methods which enable chaotic behaviour in noisy systems to be determined are presented. |

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### Contents

Preface | 1 |

Noisy Dynamics Systems | 7 |

FokkerPlanckKolmogorov Equation | 17 |

Chaotic and Regular Stochastic Processes | 30 |

Poincare Maps for Noisy Systems | 38 |

Random Lyapunov Exponents | 54 |

Examples | 68 |

Stochastic Sensitivity Functions and Chaos | 89 |

Appendix | 116 |

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amplitude approximation Argoul bifurcation diagram calculated Cantor set chaotic stochastic process characteristic for chaotic const constant Coullet Crutchfield Definition described deterministic system different values differential equations elementary events equa example existence following form formula Herzel initial condition initial value problem interval kind region Kolmogorov entropy Lett logistic map Lyapunov exponents distribution Manneville Markovian process Math maxima maximum Lyapunov exponent maximum one-dimensional Lyapunov mean Poincare map mean value neighbourhood noise variance noisy dynamics system noisy system orbit Oseledec parameters periodic phase space phase trajectory Phys Physica probability density function process with bifurcation process x(t,co Prog random noise random variables realizations regular stochastic process response route to chaos shown in Figure shows chaotic behaviour solution of equation spectral density stochastic sensitivity function sure solution t,co t+At tangent space Theor theorem tion Tresser unstable regions values of noise variational equation versus P(x white noise zero zone