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Attractor of Smale-Williams type in an autonomous distributed system. (English) Zbl 1335.37014

Summary: 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.

MSC:

37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
37D20 Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.)
37M05 Simulation of dynamical systems
35B36 Pattern formations in context of PDEs
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References:

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