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Topaj–Pikovsky Involution in the Hamiltonian Lattice of Locally Coupled Oscillators

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Abstract

We discuss the Hamiltonian model of an oscillator lattice with local coupling. The Hamiltonian model describes localized spatial modes of nonlinear the Schrödinger equation with periodic tilted potential. The Hamiltonian system manifests reversibility of the Topaj–Pikovsky phase oscillator lattice. Furthermore, the Hamiltonian system has invariant manifolds with asymptotic dynamics exactly equivalent to the Topaj–Pikovsky model. We examine the stability of trajectories belonging to invariant manifolds by means of numerical evaluation of Lyapunov exponents. We show that there is no contradiction between asymptotic dynamics on invariant manifolds and conservation of phase volume of the Hamiltonian system. We demonstrate the complexity of dynamics with results of numerical simulations.

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Acknowledgments

The authors thank Prof. A. V. Borisov, who inspired them to study the Topaj–Pikovsky problem and its possible generalizations. The authors thank A. R. Safin for bringing the paper [34] to their attention. The authors thank Prof. A. Pikovsky, L. A. Smirnov, I. R. Sataev and A. O. Kazakov for useful discussions.

Funding

The work of V. P. Kruglov and S. P. Kuznetsov was supported by the grant of the Russian Science Foundation (project no. 15-12-20035) (formulation of the problem, analytical calculations and research of related topics (Sections 1 and 2)). The work of V. P. Kruglov was supported by the grant of the Russian Science Foundation (project no. 19-71-30012) (analytical and numerical calculations and interpretation of obtained results (Sections 3–6)).

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Authors and Affiliations

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

    Vyacheslav P. Kruglov & Sergey P. Kuznetsov

  2. Kotelnikov’s Institute of Radio-Engineering and Electronics of RAS, Saratov Branch, ul. Zelenaya 38, Saratov, 410019, Russia

    Vyacheslav P. Kruglov & Sergey P. Kuznetsov

  3. Steklov Mathematical Institute, Russian Academy of Sciences, ul. Gubkina 8, Moscow, 119991, Russia

    Vyacheslav P. Kruglov

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  1. Vyacheslav P. Kruglov
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Correspondence to Vyacheslav P. Kruglov or Sergey P. Kuznetsov.

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Kruglov, V.P., Kuznetsov, S.P. Topaj–Pikovsky Involution in the Hamiltonian Lattice of Locally Coupled Oscillators. Regul. Chaot. Dyn. 24, 725–738 (2019). https://doi.org/10.1134/S1560354719060108

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  • DOI: https://doi.org/10.1134/S1560354719060108

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