Regular and Chaotic Dynamics

, Volume 18, Issue 3, pp 214–225 | Cite as

Generalized synchronization of identical chaotic systems on the route from an independent dynamics to the complete synchrony

  • Alexey Yu. JalnineEmail author


The transition from asynchronous hyperchaos to complete synchrony in coupled identical chaotic systems may either occur directly or be mediated by a preliminary stage of generalized synchronization. In the present paper we investigate the underlying mechanisms of realization of the both scenarios. It is shown that a generalized synchronization arises when the manifold of identically synchronous states M is transversally unstable, while the local transversal contraction of phase volume first appears in the areas of phase space separated from M and being visited by the chaotic trajectories. On the other hand, a direct transition from an asynchronous hyperchaos to the complete synchronization occurs, under variation of the controlling parameter, if the transversal stability appears first on the manifold M, and only then it extends upon the neighboring phase volume. The realization of one or another scenario depends upon the choice of the coupling function. This result is valid for both unidirectionally and mutually coupled systems, that is confirmed by theoretical analysis of the discrete models and numerical simulations of the physically realistic flow systems.


synchronization chaotic dynamics strange attractors 

MSC2010 numbers

34D06 34D08 34D20 34D30 37C60 37C70 37B25 37D05 37D45 


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  1. 1.
    Fujisaka, H. and Yamada, T., Stability Theory of Synchronized Motion in Coupled-Oscillator Systems, Progr. Theoret. Phys., 1983, vol. 69, no. 1, pp. 32–47.MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Afraimovich, V. S., Verichev, N. N., and Rabinovich, M. I., Stochastic Synchronization of Oscillation in Dissipative Systems, Izv. Vyssh. Uchebn. Zaved. Radiofiz., 1986, vol. 29, no. 9, pp. 1050–1060 [Radiophys. Quantum Electron., 1986, vol. 29, pp. 795–802].MathSciNetGoogle Scholar
  3. 3.
    Pikovsky, A. S., On the Interaction of Strange Attractors, Z. Phys. B, 1984, vol. 55, no. 2, pp. 149–154.MathSciNetCrossRefGoogle Scholar
  4. 4.
    Pecora, L.M. and Carroll, T. L., Synchronization in Chaotic Systems, Phys. Rev. Lett., 1990, vol. 64, no. 8, pp. 821–824.MathSciNetCrossRefGoogle Scholar
  5. 5.
    Pecora, L.M. and Carroll, T. L., Driving Systems with Chaotic Signals, Phys. Rev. A, 1991, vol. 44, no. 4, pp. 2374–2383.CrossRefGoogle Scholar
  6. 6.
    Rulkov, N. F., Sushchik, M. M., Tsimring, L. S., and Abarbanel, H. D. I., Generalized Synchronization of Chaos in Directionally Coupled Chaotic Systems, Phys. Rev. E, 1955, vol. 51, no. 2, pp. 980–994.CrossRefGoogle Scholar
  7. 7.
    Kocarev, L. and Parlitz, U., Generalized Synchronization, Predictability and Equivalence of Unidirectionally Coupled Systems, Phys. Rev. Lett., 1996, vol. 76, no. 11, pp. 1816–1819.CrossRefGoogle Scholar
  8. 8.
    Hunt, B.R., Ott, E., and Yorke, J. A., Differentiable Generalized Synchronization of Chaos, Phys. Rev. E, 1997, vol. 55, no. 4, pp. 4029–4034.MathSciNetCrossRefGoogle Scholar
  9. 9.
    Boccaletti, S., Kurths, J., Osipov, G., Valladares, D. L., and Zhou, C. S., The Synchronization of Chaotic Systems, Phys. Rep., 2002, vol. 366, nos. 1–2, pp. 1–101.MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Brown, R. and Kocarev, L., A Unifying Definition of Synchronization for Dynamical Systems, Chaos, 2000, vol. 10, no. 2, pp. 344–349.MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Abarbanel, H. D. I., Rulkov, N. F., and Sushchik, M. M., Generalized Synchronization of Chaos: The Auxiliary System Approach, Phys. Rev. E, 1996, vol. 53, no. 5, pp. 4528–4535.CrossRefGoogle Scholar
  12. 12.
    Parlitz, U., Junge, L., and Kocarev, L., Nonidentical Synchronization of Identical Systems, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 1999, vol. 9, no. 12, pp. 2305–2309.CrossRefGoogle Scholar
  13. 13.
    González-Miranda, J.M., Generalized Synchronization in Directionally Coupled Systems with Identical Individual Dynamics, Phys. Rev. E, 2002, vol. 65, no. 4, 047202, 4 pp.CrossRefGoogle Scholar
  14. 14.
    Uchida, A., McAllister, R., Meucci, R., and Roy, R., Generalized Synchronization of Chaos in Identical Systems with Hidden Degrees of Freedom, Phys. Rev. Lett., 2003, vol. 91, no. 17, 174101, 4 pp.CrossRefGoogle Scholar
  15. 15.
    Pyragas, K., Weak and Strong Synchronization of Chaos, Phys. Rev. E, 1996, vol. 54, no. 5, R4508–R4511.CrossRefGoogle Scholar
  16. 16.
    Shabunin, A., Astakhov, V., and Kurths, J., Quantitative Analysis of Chaotic Synchronization by Means of Coherence, Phys. Rev. E, 2005, vol. 72, no. 1, 016218, 11 pp.MathSciNetCrossRefGoogle Scholar
  17. 17.
    Shabunin, A. V., Astakhov, V. V., Demidov, V.V., and Efimov, A.V., Multistability and Synchronization of Chaos in Maps with “Internal” Coupling, Radiotekhnika i Elektronika, 2008, vol. 53, no. 6, pp. 702–712 [J. Commun. Technol. Electr., 2008, vol. 53, no. 6, pp. 666–675].Google Scholar
  18. 18.
    Yanchuk, S., Maistrenko, Yu., and Mosekilde, E., Synchronization of Time-Continuous Chaotic Oscillators, Chaos, 2003, vol. 13, no. 1, pp. 388–400.MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Neumann, E., Sushko, I., Maistrenko, Yu., and Feudel, U., Synchronization and Desynchronization under the Influence of Quasiperiodic Forcing, Phys. Rev. E, 2003, vol. 67, no. 2, 026202, 15 pp.MathSciNetCrossRefGoogle Scholar
  20. 20.
    de Sousa Vieira, M. and Lichtenberg, A. J., Nonuniversality of Weak Synchronization in Chaotic Systems, Phys. Rev. E, 1997, vol. 56, no. 4, R3741–R3744.CrossRefGoogle Scholar
  21. 21.
    Tamasevicius, A. and Cenys, A., Synchronizing Hyperchaos with a Single Variable, Phys. Rev. E, 1997, vol. 55, no. 1, pp. 297–299.CrossRefGoogle Scholar
  22. 22.
    Gauthier, D. J. and Bienfang, J.C., Intermittent Loss of Synchronization in Coupled Chaotic Oscillators: Toward a New Criterion for High-Quality Synchronization, Phys. Rev. Lett., 1996, vol. 77, no. 9, pp. 1751–1754.CrossRefGoogle Scholar
  23. 23.
    Junge, L. and Parlitz, U., Synchronization Using Dynamic Coupling, Phys. Rev. E, 2001, vol. 64, no. 5, 055204(R), 4 pp.CrossRefGoogle Scholar
  24. 24.
    Pikovsky, A. S. and Grassberger, P., Symmetry Breaking Bifurcation for Coupled Chaotic Attractors, J. Phys. A, 1991, vol. 24, no. 19, pp. 4587–4597.MathSciNetCrossRefGoogle Scholar
  25. 25.
    Ding, M. and Yang, W., Observation of Intermingled Basins in Coupled Oscillators Exhibiting Synchronized Chaos, Phys. Rev. E, 1996, vol. 54, no. 3, pp. 2489–2494.MathSciNetCrossRefGoogle Scholar
  26. 26.
    Maistrenko, Yu. L., Maistrenko, V. L., Popovich, A., and Mosekilde, E., Transverse Instability and Riddled Basins in a System of Two Coupled Logistic Maps, Phys. Rev. E, 1998, vol. 57, no. 3, pp. 2713–2724.MathSciNetCrossRefGoogle Scholar
  27. 27.
    Kuznetsov, S.P., Example of a Physical System with a Hyperbolic Attractor of the Smale-Williams Type, Phys. Rev. Lett., 2005, vol. 95, 144101, 4 pp.CrossRefGoogle Scholar
  28. 28.
    Kuznetsov, S.P. and Seleznev, E.P., A Strange Attractor of the Smale-Williams Type in the Chaotic Dynamics of a Physical System, Zh. Eksper. Teoret. Fiz., 2006, vol. 129, no. 2, pp. 400–412 [J. Exp. Theor. Phys., 2006, vol. 102, no. 2, pp. 355–364].MathSciNetGoogle Scholar

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© Pleiades Publishing, Ltd. 2013

Authors and Affiliations

  1. 1.Saratov Branch of Kotel’nikov’s Institute of Radio-Engineering and Electronics of RASSaratovRussia

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