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


For citation:

Kuznetsov A. P., Turukina L. V. Stable quasi-periodic and periodic regimes initiated by the short pulses in system with unstable limit cycle. Izvestiya VUZ. Applied Nonlinear Dynamics, 2006, vol. 14, iss. 1, pp. 72-81. DOI: 10.18500/0869-6632-2006-14-1-72-81

This is an open access article distributed under the terms of Creative Commons Attribution 4.0 International License (CC-BY 4.0).
Full text PDF(Ru):
(downloads: 119)
Language: 
Russian
Article type: 
Article
UDC: 
517.9

Stable quasi-periodic and periodic regimes initiated by the short pulses in system with unstable limit cycle

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

The dynamics of a system with unstable limit cycle under the periodic sequence of delta-pulses is considered. It is shown, that stable quasi-periodic regimes and phase lock regimes (synchronization) are observed within a narrow range of parameters of the external action in the system with cubic nonlinearity. Influence of main system’s parameters to the stable quasi-periodic regimes and phase lock regimes is investigated.

Key words: 
Reference: 
  1. Neymark YuI, Landa PS. Stochastic and chaotic fluctuations. Moscow: Nauka; 1987. 422 p. (In Russian).
  2. Berger P, Pomo I, Vidal K. Order in chaos. Moscow: Mir; 1991. 366 p. (In Russian).
  3. Ott E. Chaos in dynamical systems. Cambridge: Cambridge university press; 1993. 385 p.
  4. Anishchenko VS. Complex fluctuations in simple systems. Moscow: Nauka; 1990. (In Russian).
  5. Winfree AT. The Geometry of Biological Time. Berlin: Springer; 1980. 779 p.
  6. Pikovsky A, Rosenblum M, Kurts J. Synchronization: A Universal Concept in Nonlinear Science. Cambridge University Press; 2001.
  7. Caldas IL, Tasson H. Limit cycles of periodically forced oscillations. Phys. Lett. 1989;A135:264–266.
  8. Steeb WH, Kunick A. Chaos in limit-cycle systems with external periodic excitation. Int. J. of Nonlinear Mechanics. 1987;22:349–361.
  9. Gonzalez DL, Piro O. Chaos in a nonlinear driven oscillator with exact solution. Phys. Rev. Lett. 1983;50(12):870–872. DOI: 10.1103/PhysRevLett.50.870.
  10. Ding EJ. Analytic treatment of periodic orbit systematics for a nonlinear driven oscillator. Phys Rev A Gen Phys. 1986;34(4):3547–3550. DOI: 10.1103/physreva.34.3547.
  11. Ding EJ. Analytic treatment of a driven oscillator with a limit cycle. Phys Rev A Gen Phys. 1987;35(6):2669–2683. DOI: 10.1103/physreva.35.2669.
  12. Ding EJ. Structure of parameter space for a prototype nonlinear oscillator. Phys. Rev. 1987;36(3):1488–1491. DOI: 10.1103/PHYSREVA.36.1488.
  13. Ding EJ. Structure of the parameter space for the van der Pol oscillator. Physica Scripta. 1988;38(1):9–16. DOI: 10.1088/0031-8949/38/1/001.
  14. Ullmann K, Caldas IL. Transitions in the parameter space of a periodically forced dissipative system. Chaos, Solitons & Fractals. 1996;7(11):1913–1921. DOI: 10.1016/S0960-0779(96)00019-7.
  15. Keener JP, Glass L. Global bifurcations of a periodically forced nonlinear oscillator. J Math Biol. 1984;21(2):175–190. DOI: 10.1007/BF00277669.
  16. Glass L, Sun J. Periodic forcing of a limit-cycle oscillator: Fixed points, Arnold tongues, and the global organization of bifurcations. Phys Rev E Stat Phys Plasmas Fluids Relat Interdiscip Topics. 1994;50(6):5077–5084. DOI: 10.1103/physreve.50.5077.
  17. Ding EJ, Hemmer PC. Exact treatment of mode locking for a piecewise linear map. Journal of Statistical Physics. 1987;46(1-2):99–110. DOI: 10.1007/BF01010333.
  18. Kuznetsov AP, Turukina LV. Kicked Van Der Pol Oscillator: From Differential Equation To Maps. Izvestiya VUZ. Applied Nonlinear Dynamics. 2001;9(6):69–82.
  19. Glass L. et al. Global bifurcations of a periodically forced biological oscillator. Phys. Rev. A. 1983;29:1348–1357. DOI: 10.1103/PHYSREVA.29.1348.
  20. Kuznetsov AP, Turukina LV. Synchronization of self-oscillating van der pol - duffing system by the short pulses. Izvestiya VUZ. Applied Nonlinear Dynamics. 2004;12(5):16–31.
  21. Kuznetsov AP, Kuznetsov SP, Ryskin NM. Nonlinear oscillations. Moscow: Fizmatlit; 2002. 292 p. (In Russian).
  22. Kuznetsov SP. Dynamic chaos. Moscow: Fizmatlit; 2001. 296 p. (In Russian).
Received: 
24.03.2005
Accepted: 
14.10.2005
Published: 
28.04.2006
Short text (in English):
(downloads: 80)