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

For citation:

Kuznetsov S. P., Turukina L. V. Attractors of Smale–Williams type in periodically kicked model systems. Izvestiya VUZ. Applied Nonlinear Dynamics, 2010, vol. 18, iss. 5, pp. 81-90. DOI: 10.18500/0869-6632-2010-18-5-81-90

This is an open access article distributed under the terms of Creative Commons Attribution 4.0 International License (CC-BY 4.0).
Full text:
(downloads: 183)
Article type: 

Attractors of Smale–Williams type in periodically kicked model systems

Kuznetsov Sergey Petrovich, Saratov Branch of Kotel`nikov Institute of Radiophysics and Electronics of Russian Academy of Sciences
Turukina L. V., Saratov State University

Examples of model non­autonomous systems are constructed and studied possessing hyperbolic attractors of Smale–Williams type in their stroboscopic maps. The dynamics is determined by application of a periodic sequence of kicks, in such way that on one period of the external driving the angular coordinate, or the phase of oscillations, behaves in accordance with an expanding circle map with chaotic dynamics.

  1. Sinai YaG. Stochasticity of Dynamic Systems. In: Gaponov AV. Nonlinear Waves. Moscow: Nauka; 1979. 192 p. (in Russian).
  2. Guckenheimer J, Holmes P. Nonlinear oscillations, dynamical systems and bifurcations of vector fields. Moscow-Izhevsk: Institute of Computer Research; 2002. 559 p. (in Russian).
  3. Devaney RL. An Introduction to Chaotic Dynamical Systems. NY: Addison-Wesley; 1989.
  4. Shilnikov L. Mathematical problems of nonlinear dynamics: a tutorial. Int. J. of Bif. and Chaos. 1997;7(9):1953–2001. DOI: 10.1142/S0218127497001527.
  5. Katok A, Hasselblatt B. Introduction to the Modern Theory of Dynamical Systems. Trans. from English. Moscow: Factorial; 1999. (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.
  7. Ott E. Chaos in Dynamical Systems. Cambridge: Cambridge University Press; 1993.
  8. Anishchenko VS, Astakhov VV, Vadivasova TE, Neiman AB, Strelkova GI, Schimansky-Geier L. Nonlinear Effects in Chaotic and Stochastic Systems. Izhevsk: Institute of Computer Sciences; 2003.
  9. 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.
  10. Kuznetsov SP. Example of a physical system with a hyperbolic attractor of a Smale–Williams type. Phys. Rev. Lett. 2005;95:144101. DOI: 10.1103/PhysRevLett.95.144101.
  11. 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.
  12. Kuznetsov AP, Sataev IR. Verification of hyperbolicity conditions for a chaotic attractor in a system of coupled nonautonomous van der Pol oscillators. Izvestiya VUZ. Applied Nonlinear Dynamics. 2006;14(5):3–29 (in Russian). DOI: 10.18500/0869-6632-2006-14-5-3-29.
  13. Kuptsov PV, Kuznetsov SP. Transition to a synchronous chaos regime in a system of coupled non-autonomous oscillators presented in terms of amplitude equations. Nelin. Dinam. 2006;2(3):307–331 (in Russian).
  14. Kuznetsov SP, Pikovsky AS. Autonomous coupled oscillators with hyperbolic strange attractors. Physica D. 2007;232(2):87–102. DOI: 10.1016/j.physd.2007.05.008.
  15. Kuznetsov AP, Kuznetsov SP, Pikovsky AS, Turukina LV. Chaotic dynamics in the systems of coupling nonautonomous oscillators with resonance and nonresonance communicator of the signal. Izvestiya VUZ. Applied Nonlinear Dynamics. 2007;15(6):75–85 (in Russian). DOI: 10.18500/0869-6632-2007-15-6-75-85.
  16. Kuznetsov SP, Ponomarenko VI. Realization of a strange attractor of the Smale–Williams type in a radiotechnical delay-fedback oscillator. Technical Physics Letters. 2008;34(9):771–773.
  17. Kuznetsov SP, Pikovsky AS. Hyperbolic chaos in the phase dynamics of a Q-switched oscillator with delayed nonlinear feedbacks. Europhysics Letters. 2008;28:10013. DOI: 10.1209/0295-5075/84/10013.
  18. Heagy JF. A physical interpretation of the Henon map. Physica D. 1992;57:436–446. DOI: 10.1016/0167-2789(92)90012-C.
  19. Kuznetsov SP. Dynamic Chaos. Moscow: Fizmatlit; 2006. 356 p. (in Russian).
  20. Benettin G, Galgani L, Giorgilli A, Strelcyn JM. Lyapunov characteristic exponents for smooth dynamical systems and for Hamiltonian systems: A method for computing all of them. Part I: Theory. Meccanica. 1980;15:9–20. Part II: Numerical application. Meccanica. 1980;15:21–30. DOI: 10.1007/BF02128237.
Short text (in English):
(downloads: 54)