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Arzhanukhina D. S., Kuznetsov S. P. System of three non-autonomous oscillators with hyperbolic chaos. The model with DA-attractor. Izvestiya VUZ. Applied Nonlinear Dynamics, 2013, vol. 21, iss. 2, pp. 163-172. DOI: 10.18500/0869-6632-2013-21-2-163-172

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System of three non-autonomous oscillators with hyperbolic chaos. The model with DA-attractor

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

We consider a system of three coupled non-autonomous van der Pol oscillators, in which the behavior of the phases over a characteridtic period is described approximately by the Fibonacci map with modification of the «Smale surgery», which leads to the appearance of DA-attractor («Derived from Anosov»). According to the numerical results, the attractor of the stroboscopic map is placed approximately on a two-dimensional torus embedded in the six-dimensional phase space and has transverse Cantor-like structure typical for this kind of attractrors.

  1. Afraimovich V, Hsu S-B. Lectures on chaotic dynamical systems. AMS/IP Studies in Advanced Mathematics, Vol. 28. American Mathematical Society, Provi-dence RI, International Press, Somerville, MA, 2003. 353 p.
  2. Hookenheimer J, Holmes P. Nonlinear oscillations, dynamic systems, and vector field bifurcations. Moscow-Izhevsk: ICR; 2002. 559 p. (In Russian).
  3. Devaney RL. An Introduction to Chaotic Dynamical Systems. New York: Addison–Wesley; 1989. 336 p.
  4. Shilnikov L. Mathematical problems of nonlinear dynamics: a tutorial. Int. J. of Bif. & Chaos. 1997;7(9):1953–2001.
  5. Kuznetsov SP. Hyperbolic strange attractors of physically realizable systems. Izvestiya VUZ. Applied Nonlinear Dynamics. 2009;17(4):5-34. DOI: 10.18500/0869-6632-2009-17-4-5-34.
  6. Kuznetsov SP. An example of a non-autonomous continuous-time system with attractor of Plykin type in the Poincaré map. Nelin. Dinam. 2009;5(3):403–424.
  7. Kuznetsov SP. Dynamical chaos and uniformly hyperbolic attractors: from mathematics to physics. Phys. Usp. 2011;54(2):119–144. DOI: 10.3367/UFNr.0181.201102a.0121.
  8. Kuznetsov SP, Seleznev EP. A strange attractor of the Smale-Williams type in the chaotic dynamics of a physical system. Journal of Experimental and Theoretical Physics. 2006;102(2):355-364. DOI 10.1134/S1063776106020166.
  9. Katok AB, Hasselblatt B. Introduction to the Modern Theory of Dynamical Systems. Cambridge: Cambridge University Press; 1999. 768 p.
  10. Coudene Y. Pictures of hyperbolic dynamical systems. Notices of the American Mathematical Society. 2006;53(1):8–13.
  11. Kuznetsov SP. Dynamic chaos. Moscow: Fizmatlit; 2001. 296 p. (In Russian).
  12. Berger P, Pomo I, Vidal K. Order in chaos. About the deterministic approach to turbulence. Moscow: Mir; 1991. 368 с. (In Russian).
  13. Schuster G. Deterministic chaos. Moscow: Mir; 1988. 240 p. (In Russian).
  14. Arzhanuhina DS, Kuznetsov SP. 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;20(6):56-66. DOI: 10.18500/0869-6632-2012-20-6-56-66.
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