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Gulaj A. P., Buh A. V. The study of multistability and external synchronization in nonautonomous system of two coupled van der pol oscillators with repulsive coupling. Izvestiya VUZ. Applied Nonlinear Dynamics, 2014, vol. 22, iss. 6, pp. 94-103. DOI:


The study of multistability and external synchronization in nonautonomous system of two coupled van der pol oscillators with repulsive coupling

Gulaj Artem Petrovich, Saratov State University
Buh Andrej Vladimirovich, Saratov State University

  In this paper we study the bifurcational mechanisms of synchronization and multistability formation in a system of two interacting van der Pol oscillators, one of which is under external harmonic forcing. We draw a two­parametric bifurcation diagram for phasereduced system and study its evolution in transition from symmetrical to asymmetrical repulsive interaction. Relying on the results of bifurcation analysis of non­reduced system we conclude that the synchronization scenarios found in the phase­reduced system correspond to the ones in the non­reduced system.


1. Pikovsky A., Rosenblum M., Kurths J. Synchronization: A Universal Concept in Nonlinear Science. Cambridge: Cambridge University Press, 2003. 2. Anishchenko V.S., Astakhov V.V., Neiman A.B. et al. Nonlinear Dynamics of Chaotic and Stochastic Systems. Tutorial and Modern Development. Berlin: Springer, 2007. 3. Balanov A.G., Janson N.B., Postnov D.E., Sosnovtseva O.V. Synchronization: From Simple to Complex. Berlin: Springer, 2009. 4. Izhikevich E.M. Weakly connected quasi-periodic oscillators, FM interactions, and multiplexing in the brain // SIAM Journal on Applied Mathematics. 1999. Vol. 59, No 6. P. 2193. 5. Anishchenko V., Nikolaev S., Kurths J. Peculiarities of synchronization of a resonant limit cycle on a two-dimensional torus // Phys. Rev. E. 2007. Oct. Vol. 76. P. 046216. 6. Anishchenko V., Nikolaev S., Kurths J. Bifurcational mechanisms of synchronization of a resonant limit cycle on a two-dimensional torus // Chaos. 2008. Vol. 18, No 3. P. 037123. 7. Anishchenko V., Astakhov S., Vadivasova T. Phase dynamics of two coupled oscillators under external periodic force // Europhysics Letters. 2009. May. Vol.86. P.30003. 8. Fujisaka H., Yamada T. Stability theory of synchronized motion in coupled-oscillator systems // Progress of Theoretical Physics. 1983. Vol. 69, No 1. P. 32. 9. Pecora L.M., Carroll T.L. Synchronization in chaotic systems // Phys. Rev. Lett. 1990. Feb. Vol. 64. P. 821. 10. Anishchenko V.S., Vadivasova T.E., Postnov D.E., Safonova M.A. Synchronization of chaos // International Journal of Bifurcation and Chaos. 1992. Vol. 2, No 3. P. 633. 11. Rosenblum M.G., Pikovsky A.S., Kurths J. Phase synchronization of chaotic oscillators // Phys. Rev. Lett. 1996. Mar. Vol. 76. P. 1804. 12. Strogatz S.H. Exploring complex networks // Nature. 2001. Vol. 410, No6825. P. 268. 13. Lu J., Zhong J., Tang Y. et al. Synchronization in output-coupled temporal Boolean networks // Sci. Rep. 2014. 09. Vol. 4. 14. Koseska A., Volkov E., Kurths J. Oscillation quenching mechanisms: Amplitude vs. oscillation death // Physics Reports. 2013. 10. Vol. 531, No 4. P. 173. 15. Кузнецов А.П., Сатаев И.Р., Тюрюкина Л.В. Вынужденная синхронизация двух  связанных автоколебательных осцилляторов ван дер Поля // Нелинейная динамика. 2011. Т. 7, No 3. С. 411. 16. Анищенко В.С., Астахов С.В., Вадивасова Т.Е., Феоктистов А.В. Численное и экспериментальное исследование внешней синхронизации двухчастотных колебаний // Нелинейная динамика. 2009. Т. 5, No 2. С. 237. 17. Astakhov S.V., Fujiwara N., Gulay A.P. et al. Hopf bifurcation and multistability in a system of phase oscillators // Phys. Rev. E. 2013. Sep. Vol. 88. P. 032908. 18. ?bard/xpp/xpp.html.

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