By Sergey P. Kuznetsov
"Hyperbolic Chaos: A Physicist’s View” offers fresh development on uniformly hyperbolic attractors in dynamical platforms from a actual instead of mathematical viewpoint (e.g. the Plykin attractor, the Smale – Williams solenoid). The structurally good attractors appear robust stochastic homes, yet are insensitive to version of capabilities and parameters within the dynamical structures. in line with those features of hyperbolic chaos, this monograph exhibits how to define hyperbolic chaotic attractors in actual structures and the way to layout a actual structures that own hyperbolic chaos.
This publication is designed as a reference paintings for college professors and researchers within the fields of physics, mechanics, and engineering.
Dr. Sergey P. Kuznetsov is a professor on the division of Nonlinear procedures, Saratov kingdom collage, Russia.
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Hyperbolic Chaos: A Physicist’s View
"Hyperbolic Chaos: A Physicist’s View” offers fresh growth on uniformly hyperbolic attractors in dynamical structures from a actual instead of mathematical standpoint (e. g. the Plykin attractor, the Smale – Williams solenoid). The structurally sturdy attractors take place robust stochastic houses, yet are insensitive to edition of features and parameters within the dynamical platforms.
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Additional resources for Hyperbolic Chaos: A Physicist’s View
Example text
Ergodic Problems in Classical Mechanics. Benjamin, New York (1968). : Lectures on Lyapunov exponents and smooth ergodic theory. In: Smooth Ergodic Theory and Its Applications. Proceedings of Symposia in Pure Mathematics, pp. 3–90. AMS (2001). : Thermodynamics of Chaotic Systems: An Introduction. Cambridge University Press, Cambridge (1993). : Lyapunov characteristic exponents for smooth dynamical systems and for Hamiltonian systems: A method for computing all of them. Meccanica 15, 9–30 (1980).
For uniformly hyperbolic attractors a measure defined in such a way for any typical orbit relating to the attractor appears to be one and the same, and is called a measure of Sinai-Ruelle-Bowen. e. along the unstable manifolds. Although across the structure of the fibers the distribution is of singular nature, it is described by a continuous function of coordinate measured along each fiber. In other words, the phase fluid substance is distributed on the fibers in a smooth manner, without pronounced local concentrations.
For concrete attractors, these sequences may be subjected to certain restrictions, “the rules of grammar”, which may be visualized by means of respective graph with a finite number of vertices and directed edges. The graph is represented formally by an adjacency matrix whose elements are zeros and ones. A unit at the position (i, j) indicates the presence of a directed edge connecting the i-th and the j-th vertices of the graph, and zero means the absence of such an allowed transition. 14 illustrates the Markov partitions for the Smale-Williams attractor and for the attractor of Plykin type.