Location via proxy:   [ UP ]  
[Report a bug]   [Manage cookies]                
Skip to main content
Springer Nature Link
Log in

Complex projective synchronization in coupled chaotic complex dynamical systems

  • Original Paper
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

    We’re sorry, something doesn't seem to be working properly.

    Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

Abstract

In previous papers, the projective factors are always chosen as real numbers, real matrices, or even real-valued functions, which means the coupled systems evolve in the same or inverse direction simultaneously. However, in many practical situations, the drive-response systems may evolve in different directions with a constant intersection angle. Therefore, the projective synchronization with respect to a complex factor, called complex projective synchronization (CPS), should be taken into consideration. In this paper, based on Lyapunov stability theory, three typical chaotic complex dynamical systems are considered and the corresponding controllers are designed to achieve the complex projective synchronization. Further, an adaptive control method is adopted to design a universal controller for partially linear systems. Numerical examples are provided to show the effectiveness of the proposed method.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Pecora, L.M., Carroll, T.L.: Synchronization in chaotic systems. Phys. Rev. Lett. 64, 821–824 (1990)

    Article  MathSciNet  Google Scholar 

  2. Ott, E., Grebogi, C., Yorke, J.A.: Controlling chaos. Phys. Rev. Lett. 64, 1196–1199 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  3. Pyragas, K.: Synchronization of coupled time-delay systems: analytical estimations. Phys. Rev. E 58, 3067–3071 (1998)

    Article  Google Scholar 

  4. Boccaletti, S., Kurths, J., Osipov, G., Valladares, D.L., Zhou, C.S.: The synchronization of chaotic systems. Phys. Rep. 366, 1–101 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  5. Li, C., Liao, X., Wong, K.: Chaotic lag synchronization of coupled time-delayed systems and its applications in secure communication. Physica D 194, 187–202 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  6. Wu, X., Wang, H., Lu, H.: Hyperchaotic secure communication via generalized function projective synchronization. Nonlinear Anal., Real World Appl. 12, 1288–1299 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  7. Kwon, O.M., Park, J.H., Lee, S.M.: Secure communication based on chaotic synchronization via interval time-varying delay feedback control. Nonlinear Dyn. 63, 239–252 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  8. Chen, T.P., Liu, X.W., Lu, W.L.: Pinning complex networks by a single controller. IEEE Trans. Circuits Syst. I, Regul. Pap. 54, 1317–1326 (2007)

    Article  MathSciNet  Google Scholar 

  9. Li, K.Z., Small, M., Fu, X.C.: Generation of clusters in complex dynamical networks via pinning control. J. Phys. A, Math. Theor. 41, 505101 (2008)

    Article  MathSciNet  Google Scholar 

  10. Zhang, H.F., Li, K.Z., Fu, X.C., Wang, B.H.: An efficient control strategy of epidemic spreading on scale-free networks. Chin. Phys. Lett. 26, 068901 (2009)

    Article  Google Scholar 

  11. Feng, J.W., Yam, P., Austin, F., Chen, X.: Synchronizing the noise-perturbed Rösler hyperchaotic system via sliding mode control. Z. Naturforsch. A 66, 6–12 (2011)

    Google Scholar 

  12. Choi, Y.P., Haa, S.Y., Yunb, S.B.: Complete synchronization of Kuramoto oscillators with finite inertia. Physica D 240, 32–44 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  13. Li, X., Leung, A.C.S., Han, X., Liu, X., Chu, Y.: Complete (anti-)synchronization of chaotic systems with fully uncertain parameters by adaptive control. Nonlinear Dyn. 63, 263–275 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  14. Wang, Z., Shi, X.: Anti-synchronization of Liu system and Lorenz system with known or unknown parameters. Nonlinear Dyn. 57, 425–430 (2009)

    Article  MATH  Google Scholar 

  15. Suresh, R., Senthilkumar, D.V., Lakshmanan, M., Kurths, J.: Global phase synchronization in an array of time-delay systems. Phys. Rev. E 82, 016215 (2010)

    Article  Google Scholar 

  16. Ghosha, D., Chowdhury, A.R.: Lag and anticipatory synchronization based parameter estimation scheme in modulated time-delayed systems. Nonlinear Anal., Real World Appl. 11, 3059–3065 (2010)

    Article  MathSciNet  Google Scholar 

  17. Wang, X., Wang, M.: Projective synchronization of nonlinear-coupled spatiotemporal chaotic systems. Nonlinear Dyn. 62, 567–571 (2010)

    Article  Google Scholar 

  18. Hu, M., Xu, Z.: Adaptive feedback controller for projective synchronization. Nonlinear Anal., Real World Appl. 9, 1253–1260 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  19. Nie, H., Xie, L., Gao, J., Zhan, M.: Projective synchronization of two coupled excitable spiral waves. Chaos 21, 023107 (2011)

    Article  MathSciNet  Google Scholar 

  20. Yu, Y., Li, H.: Adaptive hybrid projective synchronization of uncertain chaotic systems based on backstepping design. Nonlinear Anal., Real World Appl. 12, 388–393 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  21. Fowler, A.C., Gibbon, J.D., McGuinness, M.J.: The complex Lorenz equations. Physica D 4, 139–163 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  22. Vladimirov, A.G., Toronov, V.Y., Derbov, V.L.: The complex Lorenz model: geometric structure, homoclinic bifurcation and one-dimensional map. Int. J. Bifurc. Chaos Appl. Sci. Eng. 8, 723–729 (1998)

    Article  MathSciNet  Google Scholar 

  23. Mahmoud, G.M., Aly, S.A., Farghaly, A.A.: On chaos synchronization of a complex two coupled dynamos system. Chaos Solitons Fractals 33, 178–187 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  24. Nian, F., Wang, X., Niu, Y., Lin, D.: Module-phase synchronization in complex dynamic system. Appl. Math. Comput. 217, 2481–2489 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  25. Mahmoud, G.M., Mahmoud, E.E.: Complete synchronization of chaotic complex nonlinear systems with uncertain parameters. Nonlinear Dyn. 62, 875–882 (2010)

    Article  MATH  Google Scholar 

  26. Mahmoud, G.M., Bountis, T., Mahmoud, E.E.: Active control and global synchronization for complex Chen and Lü systems. Int. J. Bifurc. Chaos Appl. Sci. Eng. 17, 4295–4308 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  27. Hu, M., Yang, Y., Xu, Z., Guo, L.: Hybrid projective synchronization in a chaotic complex nonlinear system. Math. Comput. Simul. 79, 449–457 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  28. Mahmoud, G.M., Bountis, T., AbdEl-Latif, G.M., Mahmoud, E.E.: Chaos synchronization of two different chaotic complex Chen and Lü systems. Nonlinear Dyn. 55, 43–53 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  29. Mahmoud, G.M., Aly, S.A., AL-Kashif, M.A.: Dynamical properties and chaos synchronization of a new chaotic complex nonlinear system. Nonlinear Dyn. 51, 171–181 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  30. Liu, S., Liu, P.: Adaptive anti-synchronization of chaotic complex nonlinear systems with unknown parameters. Nonlinear Anal., Real World Appl. (2011). doi:10.1016/j.nonrwa.2011.05.006

    Google Scholar 

  31. Sun, Y.J.: Generalized projective synchronization for a class of chaotic systems with parameter mismatching, unknown external excitation, and uncertain input nonlinearity. Commun. Nonlinear Sci. Numer. Simul. 16, 3863–3870 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  32. Ghosh, D.: Generalized projective synchronization in time-delayed systems: nonlinear observer approach. Chaos 19, 013102 (2009)

    Article  MathSciNet  Google Scholar 

  33. Cai, N., Jing, Y., Zhang, S.: Modified projective synchronization of chaotic systems with disturbances via active sliding mode control. Commun. Nonlinear Sci. Numer. Simul. 15, 1613–1620 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  34. Wen, G.: Designing Hopf limit circle to dynamical systems via modified projective synchronization. Nonlinear Dyn. 63, 387–393 (2011)

    Article  Google Scholar 

  35. Wu, Z., Fu, X.: Adaptive function projective synchronization of discrete chaotic systems with unknown parameters. Chin. Phys. Lett. 27, 050502 (2010)

    Article  Google Scholar 

  36. Du, H., Zeng, Q., Wang, C., Ling, M.: Function projective synchronization in coupled chaotic systems. Nonlinear Anal., Real World Appl. 11, 705–712 (2010)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. College of Mathematics and Information Science, Jiangxi Normal University, Nanchang, 330022, China

    Zhaoyan Wu

  2. Institute for Pure and Applied Mathematics, University of California, Los Angeles, CA, 90095-7121, USA

    Jinqiao Duan

  3. Department of Applied Mathematics, Illinois Institute of Technology, Chicago, IL, 60616, USA

    Jinqiao Duan

  4. Department of Mathematics, Shanghai University, Shanghai, 200444, China

    Xinchu Fu

Authors
  1. Zhaoyan Wu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jinqiao Duan
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Xinchu Fu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xinchu Fu.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Wu, Z., Duan, J. & Fu, X. Complex projective synchronization in coupled chaotic complex dynamical systems. Nonlinear Dyn 69, 771–779 (2012). https://doi.org/10.1007/s11071-011-0303-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-011-0303-0

Keywords

Search

Navigation