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Chaos in Co-operative Dynamics of Alternately Synchronized Ensembles of Globally Coupled Self-oscillators

  • Sergey P. Kuznetsov

Abstract

In many cases dynamics of spatially extended systems can be thought of as cooperative action of a large number of relatively simple elements, like interacting limit-cycle oscillators, and treated as low-dimensional dynamics in terms of appropriate collective modes (Cross and Hohenberg, 1993), In this context, a paradigm concept is the Kuramoto model of ensemble of globally coupled phase oscillators distributed in some range of the natural frequencies, and the synchronizationdesynchronization transition in such ensemble (Kuramolo, 1984; Strogalz, 2000; Pikovsky et al., 2002; Acebrón, 2005). Now, an interesting question arises about a possibility of occurrence of hyperbolic attractors on the level of the collective modes. This problem may be of interest in various fields, e.g. in hydrodynamics, in laser physics and nonlinear optics, electronics, neurodynamics. In this chapter we start from a short introduction to dynamics of a globally coupled ensemble of limitcycle oscillators and discuss different levels of reduction, including slow complex amplitude equations and phase equations; the last represents exactly the Kuramoto model. Then, we turn to some artificial, but feasible and analytically convenient model of two alternately synchronized and desynchronized interacting ensembles of globally coupled oscillators, which demonstrate transformation of the phase of collective excitation in the course of dynamic evolution in accordance with the expanding circle map at successive stages of synchronization of the subsystems. The consideration is based on a joint work with Pikovsky and Rosenblum (Kuznetsov et al., 2010).

Keywords

Collective Mode Collective Excitation Phase Oscillator Fast Oscillation Frequency Detuning 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. Acebrón, J.A.. Bonilla. L.L.. Pérez Vicente, C.J., Ritort, F.: The Kuramoto model: A simple paradigm for synchronization phenomena. Rev. Mod. Phys. 77, 137–185 (2005).ADSCrossRefGoogle Scholar
  2. Cross, M.C., Hohenberg, P.C.: Pattern formation outside of equilibrium. Rev. Mod. Phys. 65, 851–1112 (1993).ADSCrossRefGoogle Scholar
  3. Dmitricv, A.S., Panas, A.I. Dynamical Chaos: New Information Carriers for Communication Systems. Fizmatlit, Moscow (2002).Google Scholar
  4. Glova, A.F.: Phase locking of optically coupled lasers. Quantum Electronics 33, 283–306 (2003).ADSCrossRefGoogle Scholar
  5. Koronovskii, A.A., Moskalenko, O.I., Hramov, A.H.: On the use of chaotic synchronization for secure communication. Physics-Uspekhi 52, 1213–1238 (2009).ADSCrossRefGoogle Scholar
  6. Kuramoto, Y.: Chemical Oscillations, Waves, and Turbulence, Springer, Berlin (1984).zbMATHCrossRefGoogle Scholar
  7. Kuznetsov, S.P., Pikovsky, A., Rosenblum, M.: Collective phase chaos in the dynamics of interacting oscillator ensembles. Chaos 20, 043134 (2010).MathSciNetADSCrossRefGoogle Scholar
  8. Lukin, K.A.: Noise radar technology. Telecommunications and Radio-Engineering 16, 8–16 (2001).Google Scholar
  9. Maistrenko, Y.L., Popovych, O.V., lass, P.A.: Desynchronization and chaos in the Kuramoto model. In: Chazottes, J.-R and Fernandez, B. (eds.): Dynamics of Coupled Map Lattices and of Related Spatially Extended Systems, pp.285–306. Springer (2005).Google Scholar
  10. Nakagawa, N. and Kuramoto, Y.: Anomalous lyapunov spectrum in globally coupled oscillators. Physica D 80, 307–316 (1995).zbMATHCrossRefGoogle Scholar
  11. Osipov, G.V., Kurths, J., Zhou, C: Synchronization in Oscillatory Networks. Springer, Berlin (2007).zbMATHCrossRefGoogle Scholar
  12. Pikovsky, A., Rosenblum, M., Kurtz. J.: Synchronization: A Universal Concept in Nonlinear Sciences. Cambridge University Press, Cambridge (2002).Google Scholar
  13. Ruelle, I., Takens, E.: On the nalure of turbulence. Commun. Math. Phys. 20, 167–192 (1971).MathSciNetADSzbMATHCrossRefGoogle Scholar
  14. Strogatz, S.H.: From Kuramoto to Crawford: exploring the onset of synchronization in populations of coupled oscillators. Physica D 143, 1–20 (2000).MathSciNetADSzbMATHCrossRefGoogle Scholar
  15. Wiesenfeld, K., Swift, J.W: Averaged equations for Josephson junction series arrays. Phys. Rev. E 51, 1020–1025(1995).ADSCrossRefGoogle Scholar

Copyright information

© Higher Education Press, Beijing and Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Sergey P. Kuznetsov
    • 1
  1. 1.Kotel’nikov’s Institute of Radio-Engineering and Electronics of RASSaratov BranchSaratovRussian Federation

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