Advertisement

Remarks on Chaos Synchronizability and Synchronization

  • Ricardo Femat
  • Gualberto Solis-Perales
Part of the Lecture Notes in Control and Information Sciences book series (LNCIS, volume 378)

Complete Synchronizability

Synchronous behavior signifies that two or more systems operate or occur at the same time and space. Then, the synchronization problem is to induce the synchronous behavior in two or more dynamical systems. Thus, as a consequence, the synchronizability is defined as the property of dynamical systems such that the synchronous behavior can be induced on two or more dynamical systems. Such a problem can be understood as the interconnection of two or more chaotic systems in such manner that they behave in synchronous way. The synchronization can be performed by feedforward or feedback interconnections [1], [2]. In this sense synchronization problem is a control one; since synchronizability is related to the intrinsic properties of nonlinear systems to be synchronized, and such properties can be studied by exploiting the control theory concepts; controllability and observability.

Keywords

Vector Field Chaotic System Slave System Chaos Synchronization Master System 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Femat, R., Alvarez-Ramirez, J.: Synchronization of a class of strictly different oscillators. Phys. Lett. A 236, 307 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Ogorzalek, M.J.: Taming chaos-II: Control. IEEE Trans. Circuits and Syst. I 40, 700 (1993)zbMATHCrossRefGoogle Scholar
  3. 3.
    Brown, R., Kocarev, L.: A unifying framework of chaos synchronization for dynamical systems. Chaos 10, 344–349 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Boccaletti, S., Pecora, L.M., Pelaez, A.: Unifying framework for chaos synchronization of coupled dynamical systems. Phys. Rev. E 63, 66219 (2001)CrossRefGoogle Scholar
  5. 5.
    Femat, R., Solís-Perales, G.: On the chaos synchronization phenomena. Phys. Lett. A 262, 183 (1999)CrossRefGoogle Scholar
  6. 6.
    Solís-Perales, G.: Sincronización de Marcha de Polípodos, Ms.Sc. Thesis, UASLP, México (in Spanish) (1999)Google Scholar
  7. 7.
    Buono, P.L., Golubitsky, M.: Models for central pattern generators for quadruped locomotion I. J. Math. Biol. 42, 291 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Pasemann, F.: Synchronized chaos and other coherent states for two coupled neurons. Phys. D 128, 236 (1999)zbMATHCrossRefGoogle Scholar
  9. 9.
    Pérez, G., Cerdeira, H.A.: Extracting mesages masked by chaos. Phys. Rev. Lett. 74, 1970 (1995)CrossRefGoogle Scholar
  10. 10.
    Wu, C.W., Chua, L.O.: A simple way to synchronize chaotic systems with application to secure communication. Int. Jour. Bifur. and Chaos 3, 1619 (1993)zbMATHCrossRefGoogle Scholar
  11. 11.
    Femat, R., Jauregi-Ortíz, R., Solís-Perales, G.: A chaos-based communication scheme via robust asymptotic feedback. IEEE Trans. Circuits and Syst. I 48, 1161 (2001)CrossRefGoogle Scholar
  12. 12.
    Rosenblum, M.G., Pikovsky, A.S., Kurths, J.: Phase synchronization of chaotic oscillators. Phys. Rev. Lett. 76, 1804 (1996)CrossRefGoogle Scholar
  13. 13.
    Rulkov, N., Sushchik, M.M., Tsimring, L.S., Abarbanel, H.D.I.: Generalized synchronization of chaos in directly coupled chaotic systems. Phys. Rev. E 51, 980 (1995)CrossRefGoogle Scholar
  14. 14.
    Mainieri, R., Rehacek, J.: Projective synchronization in three-dimensional chaotic systems. Phys. Rev. Lett. 82, 3042 (1999)CrossRefGoogle Scholar
  15. 15.
    Udwadia, F.E., Raju, N.: Some global properties of a pair of coupled maps: Quasi-symmetry, periodicity and synchronicity. Physica D 58, 347 (1998)MathSciNetGoogle Scholar
  16. 16.
    Josić, K.: Synchronization of chaotic systems and invariant manifolds. Nonlinearity 13, 1321 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Hermann, R., Krener, A.J.: Nonlinear controlability and observability. IEEE Trans. Automat. Control 22, 728 (1977)zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Isidori, A.: Nonlinear Control Systems. Springer, Berlin (1989)zbMATHGoogle Scholar
  19. 19.
    Nijmeijer, H., van der Schaft, A.: Nonlinear Dynamical Control Systems. Springer, New York (1990)zbMATHGoogle Scholar
  20. 20.
    Kailath, T.: Linear Systems. Prentice-Hall, Englewood Clifs (1980)zbMATHGoogle Scholar
  21. 21.
    Holstein-Rathlou, N.H., Yip, K.P., Sosnovtseva, O.V., Mosekilde, E.: Synchronization phenomena in nephron-nephron interaction. Chaos 11, 417 (2001)CrossRefGoogle Scholar
  22. 22.
    Kocarev, L., Parlitz, U.: General approach for chaotic synchronization with applications to communications. Phys. Rev. Letts. 74, 5028 (1995)CrossRefGoogle Scholar
  23. 23.
    Xiaofeng, G., Lai, C.H.: On the synchronization of different chaotic oscillations. Chaos Solitons and Fractals 11, 1231–1235 (2000)zbMATHCrossRefGoogle Scholar
  24. 24.
    Femat, R., Solís-Perales, G.: Synchronization of chaotic systems with different order. Phys. Rev. E 65, 036226 (2002)CrossRefGoogle Scholar
  25. 25.
    Nijmeijer, H., Marels, M.Y.: An observer looks at synchronization. IEEE Trans. Circuits and Syst. I 44, 307 (1997)Google Scholar
  26. 26.
    Kocarev, L., Parlitz, U., Hu, B.: Lie derivatives in dynamical systems. Chaos, Solitons and Fractals 9, 1359 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Solís-Perales, G.: Synchronization of nonlinear systems, Ph.D. Thesis, U.A.S.L.P., México (in spanish) (2002)Google Scholar
  28. 28.
    Solís-Perales, G., Ayala, V., Klieman, W., Femat, R.: Complete synchronizability of chaotic systems: a geometric approach. Chaos 13, 495 (2002)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Ricardo Femat
    • Gualberto Solis-Perales

      There are no affiliations available

      Personalised recommendations