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Vyacheslav Kruglov

ul. Astrahanskaya 83, Saratov, 410012, Russia
Saratov State University


Kuznetsov S. P., Kruglov V. P.
Verification of Hyperbolicity for Attractors of Some Mechanical Systems with Chaotic Dynamics
2016, vol. 21, no. 2, pp.  160-174
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.
Keywords: dynamical system, chaos, attractor, hyperbolic dynamics, Lyapunov exponent, Smale – Williams solenoid, parametric oscillations
Citation: Kuznetsov S. P., Kruglov V. P.,  Verification of Hyperbolicity for Attractors of Some Mechanical Systems with Chaotic Dynamics, Regular and Chaotic Dynamics, 2016, vol. 21, no. 2, pp. 160-174
Kruglov V. P., Kuznetsov S. P., Pikovsky A.
Attractor of Smale–Williams Type in an Autonomous Distributed System
2014, vol. 19, no. 4, pp.  483-494
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.
Keywords: Smale–Williams solenoid, hyperbolic attractor, chaos, Swift–Hohenberg equation, Lyapunov exponent
Citation: Kruglov V. P., Kuznetsov S. P., Pikovsky A.,  Attractor of Smale–Williams Type in an Autonomous Distributed System, Regular and Chaotic Dynamics, 2014, vol. 19, no. 4, pp. 483-494

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