ISSN 0869-6632 (Print)
ISSN 2542-1905 (Online)


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Arzhanukhina D. S., Kuznetsov S. P. System of three nonautonomous oscillators with hyperbolic chaos. Part I The model with dynamics on attractor governed by Arnold’s cat map on torus. Izvestiya VUZ. Applied Nonlinear Dynamics, 2012, vol. 20, iss. 6, pp. 56-66. DOI: 10.18500/0869-6632-2012-20-6-56-66

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Russian
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Article
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517.9

System of three nonautonomous oscillators with hyperbolic chaos. Part I The model with dynamics on attractor governed by Arnold’s cat map on torus

Autors: 
Arzhanukhina Darja Sergeevna, Saratov State University
Kuznetsov Sergey Petrovich, Saratov Branch of Kotel`nikov Institute of Radiophysics and Electronics of Russian Academy of Sciences
Abstract: 

In this paper a system of three coupled nonautonomous self-oscillatory elements is studied, in which the behavior of oscillators phases on a period of the coefficients variation in the equations corresponds to the Anosov map demonstrating chaotic dynamics. Results of numerical studies allow us to conclude that the attractor of the Poincare map can be viewed as an object roughly represented by a two-dimensional torus embedded in the sixdimensional phase space of the Poincare map, on which the dynamics is the hyperbolic ´ chaos intrinsic to Anosov’s systems.

Reference: 
  1. Kuznetsov SP. Dynamic Chaos (Course of Lectures). Moscow: Fizmatlit; 2001. 296 p. (in Russian).
  2. Berge P, Pomeau Y, Vidal C. Order Within Chaos. Wiley; 1987. 329 p.
  3. Schuster G. Deterministic Chaos. Wiley; 1995. 320 p.
  4. Lichtenberg AJ, Lieberman MA. Regular and Chaotic Dynamics. New York: Springer; 1992. 692 p. DOI: 10.1007/978-1-4757-2184-3.
  5. Anishchenko VS, Vadivasova TE, Astakhov VV. Nonlinear Dynamics of Chaotic and Stochastic Systems. Foundations and Selected Problems. Saratov: Saratov University Publishing; 1999. 368 p. (in Russian).
  6. Afraimovich V, Hsu SB. Lectures on Chaotic Dynamical Systems. AMS/IP Studies in Advanced Mathematics, Vol.28. American Mathematical Society, Providence RI, International Press, Somerville, MA; 2003. 353 p.
  7. Guckenheimer J, Holmes P. Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. New York: Springer; 1983. 462 p. DOI: 10.1007/978-1-4612-1140-2.
  8. Devaney RL. An Introduction to Chaotic Dynamical Systems. NY: Addison–Wesley; 1989. 360 p.
  9. Shilnikov L. Mathematical problems of nonlinear dynamics: A tutorial. International Journal of Bifurcation and Chaos. 1997;7(9):1353–2001. DOI: 10.1142/S0218127497001527.
  10. Kuznetsov SP. Hyperbolic strange attractors of physically realizable systems. Izvestiya VUZ. Applied Nonlinear Dynamics. 2009;17(4):5–34 (in Russian). DOI: 10.18500/0869-6632-2009-17-4-5-34.
  11. Kuznetsov SP. An example of a non-autonomous continuous-time system with attractor of Plykin-type in the Poincaré map. Russian Journal of Nonlinear Dynamics. 2009;5(3):403–424 (in Russian). DOI: 10.20537/nd0903007.
  12. Kuznetsov SP. Dynamical chaos and uniformly hyperbolic attractors: from mathematics to physics. Phys. Usp. 2011;54(2):119–144. DOI: 10.3367/UFNe.0181.201102a.0121.
  13. Kuznetsov SP, Seleznev EP. A strange attractor of the Smale-Williams type in the chaotic dynamics of a physical system. J. Exp. Theor. Phys. 2006;102(2):355–364. DOI: 10.1134/S1063776106020166.
  14. Belykh V, Belykh I, Mosekilde E. Hyperbolic Plykin attractor can exist in neuron models. International Journal of Bifurcation and Chaos. 2005;15(11):3567–3578. DOI: 10.1142/S0218127405014222.
  15. Kuznetsov SP. Plykin type attractor in electronic device simulated in MULTISIM. Chaos. 2011;21(4):043105. DOI: 10.1063/1.3646903.
  16. Isaeva OB, Jalnine AY, Kuznetsov SP. Arnold’s cat map dynamics in a system of coupled nonautonomous van der Pol oscillators. Phys. Rev. E. 2006;74(4):046207. DOI: 10.1103/PhysRevE.74.046207.
  17. Kuznetsov SP, Pikovsky A. Autonomous coupled oscillators with hyperbolic strange attractors. Physica D. 2007;232(2):87–102. DOI: 10.1016/j.physd.2007.05.008.
  18. Coudene Y. Pictures of hyperbolic dynamical systems. Notices of the American Mathematical Society. 2006;53(1):8–13.
Received: 
22.06.2012
Accepted: 
04.09.2012
Published: 
29.03.2013
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