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

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Roman J. P., Kuznetsov A. P., Turukina L. V. Dynamics of three coupled van der Pol oscillators with non-identical controlling parameters. Izvestiya VUZ. Applied Nonlinear Dynamics, 2011, vol. 19, iss. 5, pp. 76-90. DOI: 10.18500/0869-6632-2011-19-5-76-90

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Dynamics of three coupled van der Pol oscillators with non-identical controlling parameters

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

We consider the chain of three dissipatively coupled self-oscillating systems with non-identical controlling parameters. We observe situations, when coupling damps different oscillators. The structure of the frequency mismatch – coupling value parameter plane is investigated with a view to the location of oscillator death area, complete synchronization area, two- and three-frequency quasiperiodic regimes. Features, connected with non-identity in controlling parameters, are considered. A possibility of complete broadband synchronization regimes and two-frequency broadband synchronization regimes is demonstrated.

  1. Pikosky A, Rosenblum M, Kurths J. Synchronization: A Universal Concept in Nonlinear Sciences. Cambridge: Cambridge University Press; 2001. 411 p. DOI: 10.1017/CBO9780511755743.
  2. Landa PS. Nonlinear Oscillations and Waves in Dynamical Systems. Dordrecht: Springer; 1996. 544 p. DOI: 10.1007/978-94-015-8763-1.
  3. Landa PS. Self-Oscillations in Systems with a Finite Number of Degrees of Freedom. Moscow: Nauka; 1980. 360 p. (in Russian).
  4. Blekhman II. Synchronization in Nature and Technology. Moscow: Nauka; 1981. 351 p. (in Russian).
  5. 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.
  6. Rand R, 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.
  7. 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.
  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. Kuznetsov AP, Paksyutov VI. On the dynamics of two coupled van der Pol – Duffing oscillators with dissipative coupling. Izvestiya VUZ. Applied Nonlinear Dynamics. 2003;11(6):48 (in Russian).
  10. 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.
  11. Kuznetsov AP, Paksyutov VI, Roman YP. Features of synchronization in a system of coupled van der Pol oscillators that are not identical in the control parameter. Tech. Phys. Lett. 2007;33(15):15—21 (in Russian).
  12. 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. 
  13. Kuznetsov AP, Roman YP. Properties of synchronization in the systems of non-identical coupled van der Pol and van der Pol–Duffing oscillators. Broadband synchronization. Physica D. 2009;238(16):1499—1506. DOI: 10.1016/j.physd.2009.04.016.
  14. Astakhov VV, Koblensky SA, Vadivasova TE, Anischenko VS. Bifurcation analysis of the dynamics of dissipatively coupled van der Pol generators. Advances in Modern Radio Electronics. 2008;9:61 (in Russian).
  15. Astahov VV, Kobljanskij SA, Shabunin AV. Bifurcation analysis of synchronization and amplitude death in coupled generators with inertial nonlinearity. Izvestiya VUZ. Applied Nonlinear Dynamics. 2010;18(2):79—97 (in Russian). DOI: 10.18500/0869-6632-2010-18-2-79-97.
  16. Kuznetsov AP, Roman JP, Seleznev EP. Synchronization in coupled self­sustained oscillators with non­-identical parameters. Izvestiya VUZ. Applied Nonlinear Dynamics. 2010;18(2):62—78 (in Russian). DOI: 10.18500/0869-6632-2010-18-2-62-78.
  17. 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.
  18. Anischenko VS, Astakhov SV, Vadivasova TE, Feoktistov AV. Numerical and experimental study of external synchronization of two-frequency oscillations. Russian Journal of Nonlinear Dynamics. 2009;5(2):237—252 (in Russian). DOI: 10.20537/nd0902006
  19. Anishchenko VS, Astakhov VV, Vadivasova TE, Strelkova GI. Synchronization of Regular, Chaotic and Stochastic Oscillations. Moscow, Izhevsk: Institute for Computer Research; 2008. 144 p. (in Russian).
  20. 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 (in Russian). DOI: 10.18500/0869-6632-2010-18-4-17-32.
  21. Kuznetsov AP, Sataev IR, Tyuryukina LV. Synchronization and multi-frequency oscillations in the chain of phase oscillators. Russian Journal of Nonlinear Dynamics. 2010;6(4):693—717 (in Russian).
  22. Landau LD. Towards the problem of turbulence. Proc. Acad. Sci. USSR. 1944;44(8):339 (in Russian).
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