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

For citation:

Kuznetsov A. P., Stankevich N. V., Turukina L. V. Features of the synchronization picture by the pulses in the system with 3-dimensional phase space by the example of the Ressler system. Izvestiya VUZ. Applied Nonlinear Dynamics, 2006, vol. 14, iss. 6, pp. 43-53. DOI: 10.18500/0869-6632-2006-14-6-43-53

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: 92)
Article type: 

Features of the synchronization picture by the pulses in the system with 3-dimensional phase space by the example of the Ressler system

Kuznetsov Aleksandr Petrovich, Saratov Branch of Kotel`nikov Institute of Radiophysics and Electronics of Russian Academy of Sciences
Stankevich Natalija Vladimirovna, National Research University "Higher School of Economics"
Turukina L. V., Saratov State University

Features of the synchronization picture in the system which limit cycle lied in three-dimensional phase space are considered. By the example of Ressler system with the periodic sequence of δ-Functions it is shown, that the synchronization picture essentially depends on a direction of the external force. Features of the synchronization tongues are found.

Key words: 
  1. Berge P, Pomeau Y, Vidal C. L’Ordre Dans Le Chaos. Paris: Hermann; 1988. 353 p.
  2. Schuster G. Deterministic chaos. New York: Weinheim. 1988. 240 p.
  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. 312 p. (In Russian).
  5. Winfree AT. The Geometry of Biological Time. Berlin: Springer; 1980. 779 p.
  6. Caldas IL, Tasson H. Limit cycles of periodically forced oscillations. Phys. Lett, 1989;135:264–266.
  7. Steeb WH, Kunick A. Chaos in limit-cycle systems with external periodic excitation. Int. J of Nonlinear Mechanics, 1987;22(5):349–361. DOI: 10.1016/0020-7462(87)90028-X.
  8. Pikovsky AS, Rosenblum MG, Kurths J. Synchronization: a universal concept in nonlinear sciences. Cambridge: University Press; 2001. 433 p.
  9. Pikovsky AS, Rosenblum MG, Osipov GV, Kurths J. Phase synchronization of chaotic oscillators by external driving. Physica. 1997;104(3-4):219–238. DOI: 10.1016/S0167-2789(96)00301-6.
  10. 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.
  11. 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.
  12. 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.
  13. Ding EJ. Structure of parameter space for a prototype nonlinear oscillator. Phys. Rev. 1987;36(3):1488–1491. DOI: 10.1103/PHYSREVA.36.1488.
  14. 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.
  15. 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.
  16. Keener JP, Glass L. Global bifurcation of a periodically forced nonlinear oscillator. J. Math. Biology, 1984;21(2):175–190. DOI: 10.1007/BF00277669.
  17. 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.
  18. 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.
  19. Glass L. et. all. Global bifurcations of a periodically forced biological oscillator. Phys. Rev. A. 1983;29(3):1348–1357. DOI: 10.1103/PHYSREVA.29.1348.
  20. Kuznetsov AP, Turukina LV. Kicked van der pol oscillator: from differential equation to maps. Izvestiya VUZ. Applied Nonlinear Dynamics. 2001;9(6):69–82.
  21. 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.
  22. Kuznetsov SP. Dynamic chaos. Moscow: Fizmatlit; 2001. 296 p. (In Russian).
Short text (in English):
(downloads: 42)