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

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Kuznetsov A. P., Sataev I. R., Sedova Y. V., Turukina L. V. On modelling the dynamics of coupled self-oscillators using the simplest phase maps. Izvestiya VUZ. Applied Nonlinear Dynamics, 2012, vol. 20, iss. 2, pp. 112-137. DOI: 10.18500/0869-6632-2012-20-2-112-137

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On modelling the dynamics of coupled self-oscillators using the simplest phase maps

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

The problem of describing the dynamics of coupled self-oscillators using discrete time systems on the torus is considered. We discuss the methodology for constructing such maps as a simple formal models, as well as physically motivated systems. We discuss the differences between the cases of the dissipative and inertial coupling. Using the method of Lyapunov exponents charts we identify the areas of two- and three-frequency quasiperiodicity and chaos. Arrangement of the Arnold resonance web is investigated and compared for different model systems. 

  1. Pikovsky A, Rosenblum M, Kurts Yu. Synchronization: A fundamental nonlinear phenomenon. Moscow: Tehnosphera; 2003. 493 p.
  2. Landa PS. Self-Oscillation in Systems with Finite Number of Degress of Freedom. Moscow: Nauka; 1980. 360 p. (in Russian).
  3. Blekhman II. Synchronization in Science and Technology. New York: ASME Press, 1988.
  4. Balanov AG, Janson NB, Postnov DE, Sosnovtseva O. Synchronization: From simple to complex. Springer; 2009. 437 p.
  5. Guckenheimer J, Holmes P. Nonlinear oscillations, dynamical systems and bifurcations of vector fields. Applied Mathematical Sciences. New York: Springer; 1983.
  6. Kuramoto Y. Chemical oscillations, waves, and turbulence. Springer Series Synergetics. Vol. 19. Berlin: Springer; 1984. 156 p.
  7. Glass L, MacKey MC. From clocks to chaos: The rhythms of life. Princeton. New-York: Princeton Univ. Press; 1988. 248 p.
  8. Winfree A. The geometry of biological time. 2nd ed. New York: Springer; 2001. 777 p.
  9. Anishchenko VS, Astakhov SV, Vadivasova TE, Strelkova GI. Synchronization of Regular, Chaotic, and Stochastic Oscillations. Moscow-Izhevsk: Institute of Computer Investigations; 2008. 136 p. (in Russian).
  10. Kuznetsov AP, Kuznetsov SP, Ryskin NM. Nonlinear oscillations. 2nd ed. Moscow: Fizmatlit; 2005. 292 p. (in Russian).
  11. Repin BG, Dubinov AE. Phasing of three vircators simulated in terms of coupled van der Pol oscillators. Tech. Phys. 2006;51:489–494. DOI: 10.1134/S1063784206040153.
  12. Kawahara T. Coupled Van der Pol oscillators – a model of excitatory and inhibitory neural interactions. Biological Cybernetics. 1980;39(1):37–43. DOI: 10.1007/BF00336943.
  13. Crowley MF, Epstein IR. Experimental and theoretical studies of a coupled chemical oscillator: phase death, multistability and in-phase and out-of-phase entrainment. J. Phys. Chem. 1989;93(6):2496–2502. DOI: 10.1021/j100343a052.
  14. Anishchenko VS, Astakhov VV, Neiman AB, Vadivasova TE, Schimansky-Geier L. Nonlinear Dynamics of Chaotic and Stochastic Systems. Tutorial and Modern Development. Springer, Berlin, Heidelberg; 2007. 460 p.
  15. Anishchenko VS. Dynamical Chaos: Models and Experiments. Appearance Routes and Structure of Chaos in Simple Dynamical Systems. World Scientific Series on Nonlinear Science. Series A. 1995;8. 384 p.
  16. Dmitriev AS, Kislov VYa. Stochastic Oscillations in Radiophysics and Electronics. Moscow: Nauka; 1989. 280 p.(in Russian).
  17. Madan R. Chua’s circuit: A paradigm for chaos. World Scientific; 1993. 1042 p.
  18. Volkov EI, Romanov VA. Bifurcations in the system of two identical diffusively coupled Brusselators. Physica Scripta. 1995;51(1):19–28. DOI: 10.1088/0031-8949/51/1/004.
  19. Schuster G. Deterministic Chaos. An Introduction. Moscow: Mir; 1988. 240 p. (in Russian).
  20. Kuznetsov SP. Dynamic chaos. Moscow: Fizmatlit; 2006. 356 p. (in Russian).
  21. Kim S, MacKay RS, Guckenheimer J. Resonance regions for families of torus maps. Nonlinearity. 1989;2(3):391–404. DOI: 10.1088/0951-7715/2/3/001.
  22. Baesens С, Guckenheimer J, Kim S, Mackay R. Simple resonance regions of torus diffeomorphisms. Patterns and dynamics in reactive media. The IMA Volumes in Mathematics and its Applications. New York: Springer. 1991;37:1–9. DOI: 10.1007/978-1-4612-3206-3_1.
  23. 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.
  24. Anishchenko V, Astakhov S, Vadivasova T. Phase dynamics of two coupled oscillators under external periodic force. Europhysics Letters. 2009;86(3):30003. DOI:
  25. Anishchenko VS, Astakhov SV, Vadivasova TE, Feoktistov AV. Numerical and experimental study of external synchronization of two-frequency oscillations. Nelin. Dinam. 2009;5(2):237–252 (in Russian).
  26. Kuznetsov AP, Sataev IR, Turukina LV. Synchronization and multi-frequency oscillations in the chain of phase oscillators. Nelin. Dinam. 2010;6(4):693–717 (in Russian).
  27. Zaslavsky GM. Physics of chaos in Hamiltonian systems. London : Imperial College Press; 1998.
  28. Morozov AD. Resonances, Cycles and Chaos in Quasi-Conservative Systems. Izhevsk: Institute of Computer Science; 2005. 424 pp. (in Russian).
  29. Vasylenko A, Maistrenko Yu, Hasler M. Modeling phase synchronization in systems of two and three coupled oscillators. Nonlinear Oscillations. 2004;7(3):301–317. DOI: 10.1007/s11072-005-0014-x.
  30. Maistrenko V, Vasylenko A, Maistrenko Y, Mosekilde E. Phase chaos and multistability in the discrete Kuramoto model. International Journal of Bifurcation and Chaos. 2010;20(6):1811–1823.
  31. 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.
  32. 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.
  33. 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.
  34. Lee E, Cross MC. Pattern formation with trapped ions. Phys. Rev. Lett. 2011;106(14):143001. DOI: 10.1103/PhysRevLett.106.143001.
  35. 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.
  36. Maistrenko Y, Popovych O, Burylko O. Mechanism of desynchronization in the finite-dimensional Kuramoto model. Phys. Rev. Lett. 2004;93(8):084102. DOI: 10.1103/PhysRevLett.93.084102.
  37. Broer H, Simo C, Vitolo R. The Hopf-saddle-node bifurcation for fixed points of 3D-diffeomorphisms: the Arnol’d resonance web. Reprint from the Belgian Mathematical Society. 2008;15(5):769–787. DOI: 10.36045/bbms/1228486406.
  38. Galkin OG. Phase-Locking for Mathieu-Type Torus Maps. Funct. Anal. Appl. 1993;27(1):1–9. DOI: 10.1007/BF01768662.
  39. Froeschle С, Lega E, Guzzo M. Analysis of the chaotic behavior of orbits diusing along the Arnold web. Celestial Mechanics and Dynamical Astronomy. 2006;95(1–4):141-153.
  40. Guzzo M, Lega E, Froeschle С. First numerical evidence of global Arnold diffusion in quasi–integrable systems. arXiv:nlin/0407059.  
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