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Stabilization of Hyperbolic Chaos by the Pyragas Method



Author(s)

Sergey Belyakin, Arsen Dzanoev, Sergey Kuznetsov

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DOI:10.17265/2159-5291/2014.12.003

Physics Faculty, Moscow State University, Moscow 119992, Russia

For a long time it was a common opinion that hyperbolic attractors are artificial mathematical constructions . However, in the recent papers  there were proposed physically realizable systems that possess, in their phase space, the set with features that are very similar to hyperbolic type of attractors. As is known, invariant sets are called hyperbolic attractors of the dynamical system if they are closed, topologically transitive subsets, and every their trajectory possesses uniform hyperbolicity. Very familiar types of the hyperbolic attractors are Smale-Williams’ solenoid and Plykin’s attractor. Further, it is well known that chaotic systems are very sensitive to the external perturbations. This property is used for controlling nonlinear systems and chaos suppression. Thus, an important question arises: Is it possible to suppress chaos in systems with hyperbolic attractors because these attractors are structurally stable subsets? In the present contribution we study the possibility of stabilization of chaotic oscillations in systems with the Smale-Williams hyperbolic attractors by means of the Pyragas method with a delay . It is shown that by means of external perturbation the dynamical system could be controllable: the hyperbolic attractor degenerates into a periodic one.

Dynamical system, hyperbolic attractors, Pyragas method