Chaos generator with the Smale–Williams attractor based on oscillation death
Russian Journal of Nonlinear Dynamics, 2017, vol. 13, no. 3, pp. 303-315
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A nonautonomous system with a uniformly hyperbolic attractor of Smale – Williams type in
a Poincaré cross-section is proposed with generation implemented on the basis of the effect of
oscillation death. The results of a numerical study of the system are presented: iteration diagrams
for phases and portraits of the attractor in the stroboscopic Poincaré cross-section, power density
spectra, Lyapunov exponents and their dependence on parameters, and the atlas of regimes. The
hyperbolicity of the attractor is verified using the criterion of angles.
Verification of Hyperbolicity for Attractors of Some Mechanical Systems with Chaotic Dynamics
Regular and Chaotic Dynamics, 2016, vol. 21, no. 2, pp. 160-174
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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.
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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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.
Hyperbolic chaos in systems with parametrically excited patterns of standing waves
Russian Journal of Nonlinear Dynamics, 2014, vol. 10, no. 3, pp. 265-277
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We outline a possibility of implementation of Smale–Williams type attractors with different stretching factors for the angular coordinate, namely, $n=3,\,5,\,7,\,9,\,11$, for the maps describing the evolution of parametrically excited standing wave patterns on a nonlinear string over a period of modulation of pump accompanying by alternate excitation of modes with the wavelength ratios of $1:n$.
Kuznetsov S. P., Kuznetsov A. S., Kruglov V. P., Hyperbolic chaos in systems with parametrically excited patterns of standing waves, Russian Journal of Nonlinear Dynamics, 2014, vol. 10, no. 3, pp. 265-277