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

For citation:

Kuznetsov A. P., Stankevich N. V., Turukina L. V. Coupled van der pol and van der Pol–Duffing oscillators: dynamics of phase and computer simulation. Izvestiya VUZ. Applied Nonlinear Dynamics, 2008, vol. 16, iss. 4, pp. 101-136. DOI: 10.18500/0869-6632-2008-16-4-101-136

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):
PDF icon 2008no4p101.pdf
(downloads: 236)
Language: 
Russian
Heading: 
Journal in the journal
Article type: 
Article
UDC: 
517.9
DOI: 
10.18500/0869-6632-2008-16-4-101-136

Coupled van der pol and van der Pol–Duffing oscillators: dynamics of phase and computer simulation

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

Synchronization in the system of coupled nonidentical and nonisochronous van der Pol oscillators with dissipative and inertial type of coupling is discussed. Generalized Adler equation is obtained and investigated in the presence of all factors. Basic symmetry of the equation, with leads to equivalence of some physical factors, is displayed. Numerical investigation of parameters space of initial differential system is realized. Results of two methods are compared and discussed.

Key words: 
-
Reference: 
  1. Pikovsky A, Rosenblum M, and Kurths J. Synchronization: A Universal Concept in Nonlinear Sciences. Cambridge, UK: Cambridge University Press; 2001. 411 p. DOI: 10.1017/CBO9780511755743.
  2. Aronson DG, Ermentrout GB, Kopell N. Amplitude response of coupled oscillators. Physica D. 1990;41(3):403–449. DOI: 10.1016/0167-2789(90)90007-C.
  3. Rand RH, Holmes PJ. Bifurcation of periodic motions in two weakly coupled van der Pol oscillators. Int. J. Non-Linear Mechanics. 1980;15(4–5):387–399. DOI: 10.1016/0020-7462(80)90024-4.
  4. Storti DW, Rand RH. Dynamics of two strongly coupled van der Pol oscillators. Int. J. Non-Linear Mechanics. 1982;17(3):143–152. DOI: 10.1016/0020-7462(82)90014-2.
  5. Chakraborty T, Rand RH. The transition from phase locking to drift in a system of two weakly coupled van der Pol oscillators. Int. J. Non-Linear Mechanics. 1988;23(5–6):369–376. DOI: 10.1016/0020-7462(88)90034-0.
  6. Poliashenko M, McKay SR, Smith CW. Chaos and nonisochronism in weakly coupled nonlinear oscillators. Phys. Rev. A. 1991;44(6):3452–3456. DOI: 10.1103/PhysRevA.44.3452.
  7. Poliashenko M, McKay SR, Smith CW. Hysteresis of synchronous – asynchronous regimes in a system of two coupled oscillators. Phys. Rev. A. 1991;43(10):5638–5641. DOI: 10.1103/physreva.43.5638.
  8. Pastor I, Perez-Garcia VM, Encinas-Sanz F, Guerra JM. Ordered and chaotic behavior of two coupled van der Pol oscillators. Phys. Rev. E. 1993;48(1):171–182. DOI: 10.1103/PhysRevE.48.171.
  9. Camacho E, Rand RH, Howland H. Dynamics of two van der Pol oscillators coupled via a bath. Int. J. Solids Structures. 2004;41(8):2133–2143. DOI: 10.1016/j.ijsolstr.2003.11.035.
  10. Kuznetsov AP, Paksyutov VI. On the dynamics of two van der Pol – Duffing oscillators with dissipative coupling. Izvestiya VUZ. Applied Nonlinear Dynamics. 2003;11(6):48–64 (in Russian).
  11. Kuznetsov AP, Paksjutov VI. Features of the parameter plane of two nonidentical coupled Van der Pol – Duffing oscillators. Izvestiya VUZ. Applied Nonlinear Dynamics. 2005;13(4):3–19 (in Russian. DOI: 10.18500/0869-6632-2005-13-4-3-19.
  12. Ivanchenko MV, Osipov GV, Shalfeev VD, Kurths J. Synchronization of two nonscalar-coupled limit-cycle oscillators. Physica D. 2004;189(1–2):8–30. DOI: 10.1016/j.physd.2003.09.035.
  13. Poston T, Stuart I. Catastrophe Theory and Its Applications. Pitman; 1978. 479 p.
  14. Arnold VI. Catastrophe Theory. Berlin: Springer; 1984. 79 p. DOI: 10.1007/978-3-642-96799-3.
  15. Kuznetsov AP, Kuznetsov SP, Ryskin NM. Nonlinear Oscillations. Ser. Modern Theory of Vibrations and Waves. Moscow: Fizmatlit; 2006. 292 p. (in Russian).
  16. Kuznetsov SP. Dynamic Chaos. Ser. Modern Theory of Vibrations and Waves. 2nd ed. Moscow: Fizmatlit; 2006. 356 p. (in Russian).
  17. Wang D, Li C, Chow SN. Normal Forms and Bifurcation of Planar Vector Fields. Cambridge University Press; 1994. 484 p.
  18. Bullough RK, Caudrey PJ, editors. Solitons. Berlin: Springer; 1980. 392 p. DOI: 10.1007/978-3-642-81448-8.
  19. Guckenheimer J, Holmes F. Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. New York: Springer; 1983. 462 p. DOI: 10.1007/978-1-4612-1140-2.
  20. 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.
  21. Arnold VI. Wavefront evolution and the Morse equivariant lemma. In: V.I. Arnold. Selected – 60. Moscow: Fazis; 1997. P. 289 (in Russian).
  22. Kuznetsov AP, Paksjutov VI, Roman JP. Properties of synchronization in the system of nonidentical coupled van der pol and van der Pol – Duffing oscillators. Broadband synchronization. Izvestiya VUZ. Applied Nonlinear Dynamics. 2007;15(4):3–15 (in Russian). DOI: 10.18500/0869-6632-2007-15-4-3-15.
  23. Kuznetsov AP, Kuznetsov SP. Critical dynamics of coupled-map lattices at onset of chaos (review). Radiophys. Quantum Electron. 1991;34(10–12):845–868. DOI: 10.1007/BF01083617.
  24. Anishchenko VS et al. Nonlinear Dynamics of Chaotic and Stochastic Systems. Berlin Springer; 2001. 374 p.
Received: 
11.01.2008
Accepted: 
25.03.2008
Published: 
31.10.2008
Short text (in English):
PDF icon a2008no4p101.doc_.pdf
(downloads: 88)
  • 1856 reads