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

For citation:

Kuznetsov A. P., Turukina L. V. Subharmonic resonance in a system of two dissipative coupled van der Pol oscillators with external force. Izvestiya VUZ. Applied Nonlinear Dynamics, 2013, vol. 21, iss. 5, pp. 69-78. DOI: 10.18500/0869-6632-2013-21-5-69-78

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: 27)
Article type: 

Subharmonic resonance in a system of two dissipative coupled van der Pol oscillators with external force

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

The problem of the excitation of two coupled oscillators is discussed in the case of the simple subharmonic resonance between the external force and eigen-frequencies of the oscillators. The corresponded phase equation is obtained. We showed that the form of the synchronization tongue and transformation of the region of the two-, three-frequency tori by varying the parameter of the coupling between the oscillators is significantly different from the case of the main resonance. We illustrated the efficiency of the phase model by the comparison of Lyapunov’s charts plotted for the case of the original system and for the phase model.

  1. Pikovsky A, Rosenblum M, Curts Yu. Synchronization. Fundamental nonlinear phenomenon. Moscow: Tehnosphera; 2003. 494 p. (In Russian).
  2. Landa PS. Self-oscillations in systems with a finite number of degrees of freedom. Moscow: Nauka; 1980. 360 p. (In Russian).
  3. Anischenko VS, Astakhov VV, Vadivasova TE. Regular and chaotic self-oscillations. Synchronization and influence of fluctuations. Textbook-monograph. Dolgoprudny: Intellect Publishing House; 2009. 312 p. (In Russian).
  4. Mettin R, Parlitz U, Lauterborn W. Bifurcation structure of the driven van der Pol oscillator. International Journal of Bifurcation and Chaos. 1993;3(6):1529–1555. DOI: 10.1142/S0218127493001203.
  5. Noris J. The closing of Arnold tongues for periodically forced limit cycle. Non-linearity. 1993;6(6):1093–1114. DOI: 10.1088/0951-7715/6/6/017.
  6. Wang D, Li Ch, Chow Sh-K. Normal forms and bifurcations of vector fields on the plane. Moscow: MCCME; 2005. 416 p. (In Russian).
  7. Vance W, Ross J. A detailed study of forced chemical oscillator: Arnold tongues and bifurcation sets. J. Chem. Phys. 1989;91(12):7654–7670. DOI: 10.1063/1.457235.
  8. Farjas J, Herrero R, Orriols F. Experimental analysis of codimensional-2 bifurcations in a periodically-forced opto-thermal oscillator. International Journal of Bifurcation and Chaos. 1998;38(7):1413–1435.
  9. Kuznetsov AP, Milovanov SV. SUBHARMONIC RESONANCE IN VAN DER Pol SYSTEM. Izvestiya VUZ. Applied Nonlinear Dynamics. 2004;12(3):74–83.
  10. Anishchenko V, Astakhov S, Vadivasova T. Phase dynamics of two coupled oscillators under external periodic force. Europhysics Letters. 2009;86(3):30003. DOI: 10.1209/0295-5075/86/30003.
  11. Anischenko VS, Astakhov VV, Vadivasova TE, Feoktistov AV. Numerical and experimental study of external synchronization of two-frequency oscillations. Nelin. Dinam. 2009;5(2):237–252.
  12. Kuznetsov AP, Sataev IR, Turukina LV. Synchronization of quasi-periodic oscillations in coupled phase oscillators. Technical Physics Letters. 2010;36(5):478–481. DOI: 10.1134/S1063785010050263.
  13. Kuznetsov AP, Sataev IR, Turukina LV. Phase dynamics of periodically driven quasiperiodic self-vibrating oscillators. Izvestiya VUZ. Applied Nonlinear Dynamics. 2010;18(4):17–32. DOI: 10.18500/0869-6632-2010-18-4-17-32.
  14. Kuznetsov AP, Sataev IR, Turukina LV. Forced synchronization of two coupled van der Pol self-oscillators. Nonlinear Dynamics. 2011;7(3):411–425.
  15. Baesens С, Guckenheimer J, Kim S, MacKay RS. Three coupled oscillators: Mode locking, global bifurcations and toroidal chaos. Physica D. 1991;49(3):387–475. DOI: 10.1016/0167-2789(91)90155-3.
  16. Khibnik AI, Braimanc Y, Kennedyd TAB, Wiesenfeldd K. Phase model analysis of two lasers with injected field. Physica D. 1998;111(1–4):295–310. DOI: 10.1016/S0167-2789(97)80017-6.
Short text (in English):
(downloads: 47)