Vyacheslav Kruglov

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.

Computer verification of hyperbolicity is provided based on statistical analysis of the angles of intersection of stable and unstable manifolds for mechanical systems with hyperbolic attractors of Smale – Williams type: (i) a particle sliding on a plane under periodic kicks, (ii) interacting particles moving on two alternately rotating disks, and (iii) a string with parametric excitation of standing-wave patterns by a modulated pump. The examples are of interest as contributing to filling the hyperbolic theory of dynamical systems with physical content.

We consider an autonomous system of partial differential equations for a onedimensional 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.