A strange attractor of the SmaleWilliams type in the chaotic dynamics of a physical system
Abstract
A nonautonomous nonlinear system is constructed and implemented as an experimental device. As represented by a 4D stroboscopic Poincare map, the system exhibits a SmaleWilliamstype strange attractor. The system consists of two coupled van der Pol oscillators whose frequencies differ by a factor of two. The corresponding Hopf bifurcation parameters slowly vary as periodic functions of time in antiphase with one another; i.e., excitation is alternately transferred between the oscillators. The mechanisms underlying the system's chaotic dynamics and onset of chaos are qualitatively explained. A governing system of differential equations is formulated. The existence of a chaotic attractor is confirmed by numerical results. Hyperbolicity is verified numerically by performing a statistical analysis of the distribution of the angle between the stable and unstable subspaces of manifolds of the chaotic invariant set. Experimental results are in qualitative agreement with numerical predictions.
 Authors:
 Russian Academy of Sciences, Saratov Branch, Institute of Radio Engineering and Electronics (Russian Federation)
 Publication Date:
 OSTI Identifier:
 21067731
 Resource Type:
 Journal Article
 Resource Relation:
 Journal Name: Journal of Experimental and Theoretical Physics; Journal Volume: 102; Journal Issue: 2; Other Information: DOI: 10.1134/S1063776106020166; Copyright (c) 2006 Nauka/Interperiodica; Article Copyright (c) 2006 Pleiades Publishing, Inc; Country of input: International Atomic Energy Agency (IAEA)
 Country of Publication:
 United States
 Language:
 English
 Subject:
 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; ATTRACTORS; BIFURCATION; CHAOS THEORY; DIFFERENTIAL EQUATIONS; EXCITATION; NONLINEAR PROBLEMS; OSCILLATORS; PERIODICITY
Citation Formats
Kuznetsov, S. P., Email: spkuz@rambler.ru, and Seleznev, E. P.. A strange attractor of the SmaleWilliams type in the chaotic dynamics of a physical system. United States: N. p., 2006.
Web. doi:10.1134/S1063776106020166.
Kuznetsov, S. P., Email: spkuz@rambler.ru, & Seleznev, E. P.. A strange attractor of the SmaleWilliams type in the chaotic dynamics of a physical system. United States. doi:10.1134/S1063776106020166.
Kuznetsov, S. P., Email: spkuz@rambler.ru, and Seleznev, E. P.. Wed .
"A strange attractor of the SmaleWilliams type in the chaotic dynamics of a physical system". United States.
doi:10.1134/S1063776106020166.
@article{osti_21067731,
title = {A strange attractor of the SmaleWilliams type in the chaotic dynamics of a physical system},
author = {Kuznetsov, S. P., Email: spkuz@rambler.ru and Seleznev, E. P.},
abstractNote = {A nonautonomous nonlinear system is constructed and implemented as an experimental device. As represented by a 4D stroboscopic Poincare map, the system exhibits a SmaleWilliamstype strange attractor. The system consists of two coupled van der Pol oscillators whose frequencies differ by a factor of two. The corresponding Hopf bifurcation parameters slowly vary as periodic functions of time in antiphase with one another; i.e., excitation is alternately transferred between the oscillators. The mechanisms underlying the system's chaotic dynamics and onset of chaos are qualitatively explained. A governing system of differential equations is formulated. The existence of a chaotic attractor is confirmed by numerical results. Hyperbolicity is verified numerically by performing a statistical analysis of the distribution of the angle between the stable and unstable subspaces of manifolds of the chaotic invariant set. Experimental results are in qualitative agreement with numerical predictions.},
doi = {10.1134/S1063776106020166},
journal = {Journal of Experimental and Theoretical Physics},
number = 2,
volume = 102,
place = {United States},
year = {Wed Feb 15 00:00:00 EST 2006},
month = {Wed Feb 15 00:00:00 EST 2006}
}

Chaos and a strange attractor were found in timedependent solutions of the full magnetohydrodynamic equations for a Faraday disc in an applied magnetic {open_quotes}seed{close_quotes} field. These were treated by the Galerkin method, truncated at five modes. The seed field is amplified several hundredfold. There are also aperiodic {open_quotes}partial pole reversals.{close_quotes} {copyright} {ital 1997 American Institute of Physics.}

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