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Attractor of Smale - Williams type in an autonomous distributed system

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Abstract

We consider an autonomous system of partial differential equations for a one-dimensional distributed medium with periodic boundary conditions. Dynamics in time consists of alternating birth and death of patterns with spatial phases transformed from one stage of activity to another by the doubly expanding circle map. So, the attractor in the Poincaré section is uniformly hyperbolic, a kind of Smale - Williams solenoid. Finite-dimensional models are derived as ordinary differential equations for amplitudes of spatial Fourier modes (the 5D and 7D models). Correspondence of the reduced models to the original system is demonstrated numerically. Computational verification of the hyperbolicity criterion is performed for the reduced models: the distribution of angles of intersection for stable and unstable manifolds on the attractor is separated from zero, i.e., the touches are excluded. The example considered gives a partial justification for the old hopes that the chaotic behavior of autonomous distributed systems may be associated with uniformly hyperbolic attractors.

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Authors and Affiliations

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

    Vyacheslav P. Kruglov & Sergey P. Kuznetsov

  2. Department of Physics and Astronomy, University of Potsdam, Karl-Liebknecht-Str. 24/25, D-14476, Potsdam-Golm, Germany

    Vyacheslav P. Kruglov & Arkady Pikovsky

  3. Saratov Branch, Kotel’nikov’s Institute of Radio-Engineering and Electronics of RAS, ul. Zelenaya 38, Saratov, 410019, Russia

    Sergey P. Kuznetsov

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  1. Vyacheslav P. Kruglov
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  2. Sergey P. Kuznetsov
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  3. Arkady Pikovsky
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Correspondence to Vyacheslav P. Kruglov.

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Kruglov, V.P., Kuznetsov, S.P. & Pikovsky, A. Attractor of Smale - Williams type in an autonomous distributed system. Regul. Chaot. Dyn. 19, 483–494 (2014). https://doi.org/10.1134/S1560354714040042

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  • DOI: https://doi.org/10.1134/S1560354714040042

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