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

For citation:

Kuznetsov A. P., Kuznetsov S. P., Savin A. V., Savin D. V. Autooscillating system with compensated dissipation: dynamics of approximated discrete map. Izvestiya VUZ. Applied Nonlinear Dynamics, 2008, vol. 16, iss. 5, pp. 127-138. DOI: 10.18500/0869-6632-2008-16-5-127-138

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: 139)
Article type: 

Autooscillating system with compensated dissipation: dynamics of approximated discrete map

Kuznetsov Aleksandr Petrovich, Saratov Branch of Kotel`nikov Institute of Radiophysics and Electronics of Russian Academy of Sciences
Kuznetsov Sergey Petrovich, Saratov Branch of Kotel`nikov Institute of Radiophysics and Electronics of Russian Academy of Sciences
Savin Aleksej Vladimirovich, Saratov State University
Savin Dmitrij Vladimirovich, Saratov State University

The pulse-driven van der Pol oscillator with the external pulse amplitude depending on the system variables is considered. The discrete map for values of the system variables just before the pulse moment was obtained by the slow-varying-amplitude method. Further the parameter space of this map was analyzed, and the existence of the Hamiltonian critical behavior in this system was shown. The remarkable fact is that our system is the system with the dissipation depending not only on the parameter values, but on the variable values too. Also the existence of the quasi-periodicity and the synchronization near the unstable cycle was shown.

Key words: 
  1. Kuznetsov SP. Dynamic Chaos. Moscow: Fizmatlit; 2006. 356 p. (in Russian).
  2. Schuster G. Deterministic Chaos. Wiley; 1995. 320 p.
  3. Feigenbaum MJ. Quantitative universality for a class of nonlinear transformations. Journal of Statistical Physics. 1978;19(1):25–52. DOI: 10.1007/BF01020332.
  4. Lichtenberg AJ, Lieberman MA. Regular and Chaotic Dynamics. New York: Springer; 1992. 692 p. DOI: 10.1007/978-1-4757-2184-3.
  5. MacKay RS. In «Long Time Predictions in Dynamics». J. Wiley and Sons, New York; 1983. 496 p.
  6. Kuznetsov AP, Kuznetsov SP, Sataev IR. A variety of period-doubling universality classes in multiparameter analysis of transition to chaos. Physica D. 1997;109(1–2):91–112. DOI: 10.1016/S0167-2789(97)00162-0.
  7. Chen C, Gyorgyi G, and Schmidt G. Universal scaling in dissipative systems. Physical Review A. 1987;35(6):2660–2668. DOI: 10.1103/physreva.35.2660.
  8. Reick C. Universal corrections to parameter scaling in period-doubling systems: Multiple scaling and crossover. Physical Review A. 1992;45(2):777–792. DOI: 10.1103/PhysRevA.45.777.
  9. Reinout G, Quispel W. Analytical crossover results for the Feigenbaum constants: Crossover from conservative to dissipative systems. Physical Review A. 1985;31(6):3924–3928. DOI: 10.1103/PhysRevA.31.3924.
  10. Kuznetsov AP, Kuznetsov SP, Ryskin NM. Nonlinear Oscillations. Moscow: Fizmatlit; 2002. 292 p. (in Russian).
  11. Lima R, Pettini M. Suppression of chaos by resonant parametric perturbations. Physical Review A. 1990;41(2):726–733. DOI: 10.1103/physreva.41.726.
  12. Kuznetsov AP, Tyuryukina LV. Forced synchronization in a system with unstable cycle. Tech. Phys. Lett. 2003;29(4):332–333. DOI: 10.1134/1.1573307.
  13. Kuznetsov AP, Turukina LV. Stable quasi-periodic and periodic regimes initiated by the short pulses in system with unstable limit cycle. Izvestiya VUZ. Applied Nonlinear Dynamics. 2006;14(1):72–81 (in Russian). DOI: 10.18500/0869-6632-2006-14-1-72-81.
  14. Kuznetsov AP, Roman JP, Stankevich NV, Turukina LV. Pulsed synchronization and synchronization in coupled systems: new aspects of classical problem. Izvestiya VUZ. Applied Nonlinear Dynamics. 2008;16(3):88–111 (in Russian). DOI: 10.18500/0869-6632-2008-16-3-88-111.
  15. Kim SY. Bicritical behavior of period doublings in unidirectionally coupled maps. Physical Review E. 1999;59(6):6585–6592. DOI: 10.1103/physreve.59.6585.
Short text (in English):
(downloads: 83)