## Discreteness and Continuity in Problems of Chaotic DynamicsThis book presents the study of ergodic properties of so-called chaotic dynamical systems. One of the central topics is the interplay between deterministic and quasi-stochastic behaviour in chaotic dynamics and between properties of continuous dynamical systems and those of their discrete approximations. Using simple examples, the author describes the main phenomena known in chaotic dynamical systems, studying topics such as the operator approach in chaotic dynamics, stochastic stability, and the so-called coupled systems. The last two chapters are devoted to problems of numerical modeling of chaotic dynamics. |

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### Contents

Operator Approach in Chaotic Dynamics | 13 |

Random Perturbations of Dynamical Systems | 53 |

Weakly Coupled Dynamical Systems | 95 |

Phase Space Discretization in Dynamical Systems | 109 |

Ergodic Properties of Some Methods for Numerical Modeling | 139 |

157 | |

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### Common terms and phrases

approximation arbitrarily Borel set bounded variation chaotic dynamical systems completes the proof consider Const construct converge weakly coordinates corresponding coupled map lattices cycle d-dimensional defined definition denote discretized map discretized system domain dyadic map e-discretized equal estimate example exists expanding constant exponential finite dimensional finite number fixed point function h functions of bounded integer intersection interval invariant measure iterations lattice Lebesgue measure LEMMA Let f linear logistic maps Lyapunov exponent map f Markov chain Markov partition multidimensional natural numbers neighborhood obtain one-dimensional origin parallelograms parameter periodic trajectory periodic turning points Perron-Frobenius operator perturbed system phase space piecewise expanding map preimages probable trajectory proof of Theorem prove random perturbations result rotation round-off errors segment sequence smooth invariant measure special partition statement statistical probability statistical properties stochastic attractor stochastic stability subset sufficiently small transition operator transition probability densities twist map uniformly unstable var(h

### Popular passages

Page xi - Hopf's ideas were taken up by Krylov, who tried to argue that the relaxation processes in a hard sphere gas are governed by this same mechanism [15]. Eventually, proofs of some of Krylov's assertions were supplied by Sinai for the hard sphere gas [16] and by Anosov for geodesic flows on a large class of smooth manifolds [17]. " The key to the behavior of Anosov systems is that the non-vanishing tangent vectors to the transverse section of a flow at a fixed point P should be classified into stable...

Page xi - The theory of non-linear dynamical systems has taken very much a second place to the development and refinement of that of linear systems over much of this century, in spite of a great deal of early pioneering work in the field by Poincare, Birkhoff and others.

Page 160 - Freidlin, Fluctuations in dynamical systems under the action of small random perturbations, "Nauka", Moscow, 1 979; English transl., MI Freidlin and AD Wentzell, Random perturbations of dynamical systems, Springer- Verlag, Berlin, 1984.

Page xi - Krylov. and others, it was not until the late 1950s and 1960s that the field really gathered momentum. In this period...

Page 160 - M. Rychlik, Bounded variation and invariant measures, Studia Math. 76 (1983), 69-80.

Page xi - Poincare sections pointed out clearly the fractal nature of the strange attractors that underlie chaotic motions, as also did various calculations of fractal dimension from numerical data. Related aspects are the interplay of deterministic chaos and stochastic noise, and the development of methods to distinguish them in experimental data. Another subject which has become a focus of attention in recent years is the rise of chaotic behaviour in quantum systems, and the features that characterize and...