Discreteness and Continuity in Problems of Chaotic Dynamics

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American Mathematical Soc., Jan 1, 1997 - Mathematics - 161 pages
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This 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
Bibliography
157
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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...

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