## Chaos, Fractals, and Noise: Stochastic Aspects of DynamicsIn recent years there has been an explosive growth in the study of physical, biological, and economic systems that can be profitably studied using densities. Because of the general inaccessibility of the mathematical literature to the nonspecialist, little diffusion of the applicable mathematics into the study of these "chaotic" systems has taken place. This book will help bridge that gap. To show how densities arise in simple deterministic systems, the authors give a unified treatment of a variety of mathematical systems generating densities, ranging from one-dimensional discrete time transformations through continuous time systems described by integro-partial-differential equations. Examples have been drawn from many fields to illustrate the utility of the concepts and techniques presented, and the ideas in this book should thus prove useful in the study of a number of applied sciences. The authors assume that the reader has a knowledge of advanced calculus and differential equations. Basic concepts from measure theory, ergodic theory, the geometry of manifolds, partial differential equations, probability theory and Markov processes, and stochastic integrals and differential equations are introduced as needed. Physicists, chemists, and biomathematicians studying chaotic behavior will find this book of value. It will also be a useful reference or text for mathematicians and graduate students working in ergodic theory and dynamical systems. |

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### Contents

Introduction | 1 |

The Toolbox | 17 |

Markov and FrobeniusPerron Operators | 37 |

Copyright | |

12 other sections not shown

### Other editions - View all

Chaos, Fractals, and Noise: Stochastic Aspects of Dynamics Andrzej Lasota,Michael C. Mackey Limited preview - 1998 |

Chaos, Fractals, and Noise: Stochastic Aspects of Dynamics Andrzej Lasota,Michael C. Mackey Limited preview - 2013 |

### Common terms and phrases

absolutely continuous arbitrary assume asymptotically periodic asymptotically stable baker transformation behavior Boltzmann equation Borel measure bounded compact support completes the proof consequence consider constant constrictive continuous function continuous time systems Corollary defined definition denotes derivative differential equations dyadic transformation dynamical system entropy ergodic exact Example exists Figure finite measure Fokker-Planck equation formula Frobenius-Perron operator corresponding given Hille-Yosida theorem implies inequality infinitesimal operator initial condition interval iterates J—oo Jx Jx Koopman operator Lebesgue integral limit lower-bound function mapping Markov operator measurable function measure preserving measure space mixing nonnegative nonsingular obtain perturbation Pf(x Pnf(x precompact probabilistic properties Proposition prove Ptf(x r-algebra random variables random vector Remark result right-hand side satisfies semidynamical system semigroup semigroup Pt}t>o sequence solution stationary density stochastic kernel stochastic process stochastic semigroup strong convergence subset supp sweeping trajectory unique Wiener process