Hyperbolic Chaos: A Physicist’s View by Sergey P. Kuznetsov
By Sergey P. Kuznetsov
"Hyperbolic Chaos: A Physicist’s View” offers contemporary growth on uniformly hyperbolic attractors in dynamical structures from a actual instead of mathematical point of view (e.g. the Plykin attractor, the Smale – Williams solenoid). The structurally reliable attractors take place robust stochastic houses, yet are insensitive to version of capabilities and parameters within the dynamical structures. in response to 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 ebook is designed as a reference paintings for collage professors and researchers within the fields of physics, mechanics, and engineering.
Dr. Sergey P. Kuznetsov is a professor on the division of Nonlinear strategies, Saratov country collage, Russia.
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4 Content and conclusions of the hyperbolic theory 29 small variation of the system, at least the positive Lyapunov exponent also varies weakly. This assertion is confirmed by numerical data for models with uniformly hyperbolic attractors, discussed in subsequent chapters, but a mathematical result, to which in this context one could refer, is unknown to the author. 7 Invariant measure of Sinai-Ruelle-Bowen For the uniformly hyperbolic attractors there exists an absolutely continuous invariant measure of Sinai-Ruelle-Bowen (Sinai, 1972; Bowen, 1975; Ruelle, 1976).
For continuous-time systems representing suspension of diffeomorphisms with uniformly hyperbolic attractors, the question of mixing and decay of correlations requires a special analysis in each case, as it depends on the nature of the distribution for times of returns to the Poincar´e section. 10 Kolmogorov-Sinai entropy Consider a set of all “words” occurring in symbolic representations of trajectories on the attractor and containing n characters. For each i-th word one can determine a probability of its occurrence p i .
It means that we should glue the edges not directly, but with their relative rotation by the angle of 2π . 5) is not appropriate as starting point for constructing physical flow systems suspending the Smale-Williams attractor, and alternative approaches must be developed. Let us consider now an explicit example of a discrete-time system with DAattractor on a torus (Kuznetsov, 2009). 9) can be decomposed to two successively applied maps, which may be interpreted naturally as corresponding to two halves of a time step.