An example of physical system with hyperbolic attractor of Smale - Williams type

Abstract

A simple and transparent example of a non-autonomous flow system, with hyperbolic strange attractor is suggested. The system is constructed on a basis of two coupled van der Pol oscillators, the characteristic frequencies differ twice, and the parameters controlling generation in both oscillators undergo a slow periodic counter-phase variation in time. In terms of stroboscopic Poincaré section, the respective four-dimensional mapping has a hyperbolic strange attractor of Smale - Williams type. Qualitative reasoning and quantitative data of numerical computations are presented and discussed, e.g. Lyapunov exponents and their parameter dependencies. A special test for hyperbolicity based on statistical analysis of distributions of angles between stable and unstable subspaces of a chaotic trajectory has been performed. Perspectives of further comparative studies of hyperbolic and non-hyperbolic chaotic dynamics in physical aspect are outlined.

7 pages, 4 figures

References in corpus (1)

• Computing Lyapunov spectra with continuous Gram-Schmidt orthonormalization

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