Topaj–Pikovsky Involution in the Hamiltonian Lattice of Locally Coupled Oscillators
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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.
Keywordsreversibility involution Hamiltonian system Topaj–Pikovsky model phase oscillator lattice
MSC2010 numbers37J15 37C10 37C70 34D08 34C15 34C60
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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  to their attention. The authors thank Prof. A. Pikovsky, L. A. Smirnov, I. R. Sataev and A. O. Kazakov for useful discussions.
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)).
Conflict of Interest
The authors declare that they have no conflicts of interest.
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