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ISSN 2542-1905 (Online)

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Kuznetsov A. P., Emelianova Y. P., Seleznev E. P. Synchronization in coupled self­sustained oscillators with non­-identical parameters. Izvestiya VUZ. Applied Nonlinear Dynamics, 2010, vol. 18, iss. 2, pp. 62-78. DOI: 10.18500/0869-6632-2010-18-2-62-78

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Synchronization in coupled self­sustained oscillators with non­-identical parameters

Kuznetsov Aleksandr Petrovich, Saratov Branch of Kotel`nikov Institute of Radiophysics and Electronics of Russian Academy of Sciences
Emelianova Yulija Pavlovna, Saratov State University
Seleznev Evgeny Petrovich, Saratov Branch of Kotel`nikov Institute of Radiophysics and Electronics of Russian Academy of Sciences

The particular properties of dynamics are discussed for dissipatively coupled van der Pol oscillators, non-identical in values of parameters controlling the Andronov–Hopf bifurcation and nonlinear dissipation. Possibility of a special synchronization regime in an infinitively long band between oscillator death and quasiperiodic areas is shown for such system. Non-identity of parameters of nonlinear dissipation results in specific form of the boundary of the main synchronization tongue, which looks like the mirror letter S. These physical features are partly revealed by means of quasiharmonic approximation and are observed in the experiments with coupled radio-electronic generators.

  1. Pikovsky A, Rosenblum M, Kurts Yu. Synchronization: A fundamental nonlinear phenomenon. Moscow: Tehnosphera; 2003. 493 p.
  2. Aronson DG, Ermentrout GB, Kopell N. Amplitude response of coupled oscillators. Physica D. 1990;41:403–449. DOI: 10.1016/0167-2789(90)90007-C.
  3. Cohen DS, Neu JC. Interacting oscillatory chemical reactors. Bifurcation theory and applications in the scientific disciplines. Ed. Gurel O, Rossler OE. Ann. N.Y. Acad. Sci. 1979;316:332–337. DOI: 10.1111/j.1749-6632.1979.tb29478.x.
  4. Neu JC. Coupled chemical oscillators. SIAM J. appl. Math. 1979;37(2):307-315. DOI: 10.1137/0137022.
  5. Minorsky N. Nonlinear oscillators. Van Nostrand; 1962.
  6. Rand RH, Holmes PJ. Bifurcation of periodic motions in two weakly coupled van der Pol oscillators. Int. J. Non-Linear Mechanics. 1980;15:387–399. DOI: 10.1016/0020-7462(80)90024-4.
  7. 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.
  8. Chakraborty T. Bifurcation analysis of two weakly coupled van der Pol oscillators. Doctoral thesis. Cornell University; 1986.
  9. 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.
  10. Pastor-Diaz I, Lopez-Fraguas A. Dynamics of two coupled van der Pol oscillators. Phys. Rev. E. 1995;52:1480–1489. DOI: 10.1103/physreve.52.1480.
  11. Pavlidis T. Biological oscillators: The Mathematical Analysis. Academic press; 1973.
  12. Poliashenko M, McKay SR, Smith CW. Chaos and nonisochronism in weakly coupled nonlinear oscillators. Phys. Rev. A. 1991;44:3452–3456. DOI: 10.1103/physreva.44.3452.
  13. Poliashenko M, McKay SR, Smith CW. Hysteresis of synchronous – asynchronous regimes in a system of two coupled oscillators. Phys. Rev. A. 1991;43:5638–5641. DOI: 10.1103/physreva.43.5638.
  14. Ivanchenko MV, Osipov GV, Shalfeev VD, Kurths J. Synchronization of two non-scalar-coupled limit-cycle oscillators. Physica D. 2004;189(1–2):8–30. DOI: 10.1016/j.physd.2003.09.035.
  15. 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.
  16. Kuznetsov AP, Stankevich NV, Turukina LV. Coupled van der pol and van der Pol–Duffing oscillators: dynamics of phase and computer simulation. Izvestiya VUZ. Applied Nonlinear Dynamics. 2008;16(4):101–136 (in Russian). DOI: 10.18500/0869-6632-2008-16-4-101-136.
  17. Kuznetsov AP, Paksyutov VI, Roman YuP. Features of the synchronization of coupled van der Pol oscillators with nonidentical control parameters. Technical Physics Letters.  2007;33(8):636–638. DOI: 10.1134/S1063785007080032.
  18. 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 .
  19. Kuznetsov AP, Kuznetsov SP, Ryskin NM. Nonlinear oscillations. Moscow: Fizmatlit; 2002. 292 p. (in Russian).
  20. Kuznetsov SP. Dynamical Chaos: Course of Lectures. Moscow: Fizmatlit; 2001. 296 p. (in Russian).
  21. Anishchenko VS. Complex oscillations in simple systems. Moscow: Nauka; 1990. 312 p. (in Russian).
  22. Kuznetsov Yuri A. Elements of applied bifurcation theory. New York: Springer; 1998. 593 p.
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