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On some simple examples of mechanical systems with hyperbolic chaos

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Abstract

Examples of mechanical systems with hyperbolic chaos are discussed, including the Thurston–Weeks–Hunt–MacKay hinge mechanism, in which conservative Anosov dynamics is realized, and dissipative systems with Smale–Williams type attractors (a particle on a plane under periodic kicks, interacting particles sliding on two alternately rotating disks, and a string with parametric excitation by modulated pump). The examples considered in the paper are interesting from the viewpoint of filling hyperbolic theory, as a well-developed field of the mathematical theory of dynamical systems, with physical content. The results of computer tests for hyperbolicity of the systems are presented that are based on the analysis of the statistics of intersection angles of stable and unstable manifolds.

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Authors and Affiliations

  1. Udmurt State University, ul. Universitetskaya 1, Izhevsk, 426034, Russia

    S. P. Kuznetsov

  2. Saratov Branch of the Kotel’nikov Institute of Radio Engineering and Electronics, Russian Academy of Sciences, ul. Zelenaya 38, Saratov, 410019, Russia

    S. P. Kuznetsov & V. P. Kruglov

  3. Saratov State University, ul. Astrakhanskaya 83, Saratov, 410012, Russia

    S. P. Kuznetsov & V. P. Kruglov

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  2. V. P. Kruglov
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Original Russian Text © S.P. Kuznetsov, V.P. Kruglov, 2017, published in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2017, Vol. 297, pp. 232–259.

In memory of Academician Dmitry Victorovich Anosov

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Kuznetsov, S.P., Kruglov, V.P. On some simple examples of mechanical systems with hyperbolic chaos. Proc. Steklov Inst. Math. 297, 208–234 (2017). https://doi.org/10.1134/S0081543817040137

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