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Nonlinear Dynamics

, Volume 71, Issue 1–2, pp 279–290 | Cite as

Complete and generalized synchronization of chaos and hyperchaos in a coupled first-order time-delayed system

  • Tanmoy Banerjee
  • Debabrata Biswas
  • B. C. Sarkar
Original Paper

Abstract

This paper explores the synchronization scenario of coupled chaotic and hyperchaotic time delay systems that are coupled through linear, dissipative and unidirectional coupling. For the present study, we choose a prototype first-order nonlinear time delay system, which is recently reported in Banerjee et al. (Nonlinlinear. Dyn., doi: 10.1007/s11071-012-0490-3, 2012); the system shows well-characterized chaotic and hyperchaotic oscillations even for a small time delay, and also, experimental implementation of the system is easy. We show that, keeping all the system design parameters the same for the two systems, if the time delays associated with the two systems are equal, then complete synchronization occurs beyond a threshold coupling strength. On the contrary, above a certain coupling strength, generalized synchronization between two identical coupled systems occurs for the unequal time delays. We derive an estimate of the coupling strength and sufficient stability conditions for all the synchronization processes using Krasovskii–Lyapunov theory. We simulate the coupled system numerically to support the analytical results. Also, we implement the coupled system in an electronic circuit to verify all the synchronization phenomena. It is shown that the experimental results agree well with our analytical and numerical results.

Keywords

Delay dynamical system Chaos synchronization Hyperchaos Time delay electronic circuit 

Notes

Acknowledgements

One of the authors (D.B.) thankfully acknowledges the financial support provided by the University of Burdwan.

References

  1. 1.
    Banerjee, T., Biswas, D., Sarkar, B.C.: Design and analysis of a first order time-delayed chaotic system. Nonlinear Dyn. (2012). doi: 10.1007/s11071-012-0490-3. Published online 12 June 2012
  2. 2.
    Pecora, L.M., Carroll, T.L.: Synchronization in chaotic systems. Phys. Rev. Lett. 64, 821–824 (1990) MathSciNetCrossRefGoogle Scholar
  3. 3.
    Rulkov, N.F., Sushchik, M.M., Tsimring, L.S., Abarbanel, H.D.I.: Generalized synchronization of chaos in directionally coupled chaotic systems. Phys. Rev. E 51, 980–994 (1995) CrossRefGoogle Scholar
  4. 4.
    Abarbanel, H.D.I., Rulkov, N.F., Sushchik, M.M.: Generalized synchronization of chaos: the auxiliary system approach. Phys. Rev. E 53, 4528–4535 (1996) CrossRefGoogle Scholar
  5. 5.
    Cai, N., Li, W., Jing, Y.: Finite-time generalized synchronization of chaotic systems with different order. Nonlinear Dyn. 64, 385–393 (2011) MathSciNetCrossRefGoogle Scholar
  6. 6.
    Rosenblum, M.G., Pikovsky, A.S., Kurths, J.: Phase synchronization of chaotic oscillators. Phys. Rev. Lett. 76, 1804–1807 (1996) CrossRefGoogle Scholar
  7. 7.
    Rosenblum, M.G., Pikovsky, A.S., Kurths, J.: From phase to lag synchronization in coupled chaotic oscillators. Phys. Rev. Lett. 78, 4193–4197 (1997) CrossRefGoogle Scholar
  8. 8.
    Miao, Q., Tang, Y., Lu, S., Fang, J.: Lag synchronization of a class of chaotic systems with unknown parameters. Nonlinear Dyn. 57, 107–112 (2009) MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Voss, H.U.: Anticipating chaotic synchronization. Phys. Rev. E 61, 5115–5119 (2002) CrossRefGoogle Scholar
  10. 10.
    Amritkar, R.E., Gupte, N.: Synchronization of chaotic orbits: the effect of a finite time step. Phys. Rev. E 47, 3889–3895 (1993) CrossRefGoogle Scholar
  11. 11.
    Stojanovski, T., Kocarev, L., Parlitz, U.: Driving and synchronizing by chaotic impulses. Phys. Rev. E 54, 2128–2131 (1996) CrossRefGoogle Scholar
  12. 12.
    Pecora, L.M., Carroll, T.L., Johnson, G.A., Mar, D.J.: Fundamentals of synchronization in chaotic systems, concepts, and applications. Chaos 7, 4 (1997) MathSciNetCrossRefGoogle Scholar
  13. 13.
    Boccaletti, S., Kurths, J., Osipov, G., Valladares, D.L., Zhou, C.S.: The synchronization of chaotic systems. Phys. Rep. 366, 1–101 (2002) MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Peng, J.H., Ding, E.J., Ding, M., Yang, W.: Synchronizing hyperchaos with a scalar transmitted signal. Phys. Rev. Lett. 76, 904–907 (1996) CrossRefGoogle Scholar
  15. 15.
    Frasca, M., Buscarino, A., Rizzo, A., Fortuna, L., Boccaletti, S.: Synchronization of moving chaotic agents. Phys. Rev. Lett. 100, 044102 (2008) CrossRefGoogle Scholar
  16. 16.
    Mackey, M.C., Glass, L.: Oscillation and chaos in physiological control system. Science 197, 287–289 (1977) CrossRefGoogle Scholar
  17. 17.
    Ikeda, K., Daido, H., Akimoto, O.: Optical turbulence: chaotic behavior of transmitted light from a ring cavity. Phys. Rev. Lett. 45, 709–712 (1980) CrossRefGoogle Scholar
  18. 18.
    Wei, J., Yu, C.: Stability and bifurcation analysis in the cross-coupled laser model with delay. Nonlinear Dyn. 66, 29–38 (2011) MathSciNetCrossRefGoogle Scholar
  19. 19.
    Yongzhen, P., Shuping, L., Changguo, L.: Effect of delay on a predator–prey model with parasitic infection. Nonlinear Dyn. 63, 311–321 (2011) MathSciNetCrossRefGoogle Scholar
  20. 20.
    Pei, L., Wang, Q., Shi, H.: Bifurcation dynamics of the modified physiological model of artificial pancreas with insulin secretion delay. Nonlinear Dyn. 63, 417–427 (2011) MathSciNetCrossRefGoogle Scholar
  21. 21.
    Boutle, I., Taylor, R.H.S., Romer, R.A.: El Niño and the delayed action oscillator. Am. J. Phys. 75, 15–24 (2007) MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Liao, X., Guo, S., Li, C.: Stability and bifurcation analysis in tri-neuron model with time delay. Nonlinear Dyn. 49, 319–345 (2007) MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Le, L.B., Konishi, K., Hara, N.: Design and experimental verification of multiple delay feedback control for time-delay nonlinear oscillators. Nonlinear Dyn. 67, 1407–1418 (2012) zbMATHCrossRefGoogle Scholar
  24. 24.
    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) MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Ji, J.C., Hansen, C.H., Li, X.: Effect of external excitations on a nonlinear system with time delay. Nonlinear Dyn. 41, 385–402 (2005) MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Ramana Reddy, D.V., Sen, A., Johnston, G.L.: Sen:time delay induced death in coupled limit cycle oscillators. Phys. Rev. Lett. 80, 5109–5112 (1998) CrossRefGoogle Scholar
  27. 27.
    Pyragas, K.: Synchronization of coupled time-delay systems: analytical estimation. Phys. Rev. E, Stat. Nonlinear Soft Matter Phys. 58, 3067–3071 (1998) CrossRefGoogle Scholar
  28. 28.
    Zhan, M., Wang, X., Gong, X., Wei, G.W., Lai, C.H.: Complete synchronization and generalized synchronization of one-way coupled time-delay systems. Phys. Rev. E 68, 036208 (2003) CrossRefGoogle Scholar
  29. 29.
    Sahaverdiev, E.M., Shore, K.A.: Generalized synchronization in time-delayed systems. Phys. Rev. E, Stat. Nonlinear Soft Matter Phys. 71, 016201 (2005) CrossRefGoogle Scholar
  30. 30.
    Senthilkumar, D.V., Lakshmanan, M.: Transition from anticipatory to lag synchronization via complete synchronization in time-delay systems. Phys. Rev. E 71, 016211 (2005) CrossRefGoogle Scholar
  31. 31.
    Srinivasan, K., Senthilkumar, D.V., Murali, K., Lakshmanan, M., Kurths, J.: Synchronization transitions in coupled time-delay electronic circuits with a threshold nonlinearity. Chaos 21, 023119 (2011) CrossRefGoogle Scholar
  32. 32.
    Nayfeh, A.H., Balachandran, B.: Applied Nonlinear Dynamics: Analytical, Computational, and Experimental Methods. Wiley, New York (1995) zbMATHCrossRefGoogle Scholar
  33. 33.
    Perez, G., Cerdeira, H.: Extracting messages masked by chaos. Phys. Rev. Lett. 74, 1970–1973 (1995) CrossRefGoogle Scholar
  34. 34.
    Kye, W.-H., Choi, M., Kurdoglyan, M.S., Kim, C.-M., Park, Y.-J.: Synchronization of chaotic oscillators due to common delay time modulation. Phys. Rev. E 70, 046211 (2004) CrossRefGoogle Scholar
  35. 35.
    Namajunas, A., Pyragas, K., Tamaševičius, A.: An electronic analog of the Mackey–Glass system. Phys. Lett. A 201, 42–46 (1995) CrossRefGoogle Scholar
  36. 36.
    Lu, H., He, Y., He, Z.: A chaos-generator: analysis of complex dynamics of a cell equation in delayed cellular neural networks. IEEE Trans. Circuits Syst. I, Fundam. Theory Appl. 45, 178–181 (1998) MathSciNetCrossRefGoogle Scholar
  37. 37.
    Mykolaitis, G., Tamaševičius, A., Čenys, A., Bumeliene, S., Anagnostopoulos, A.N., Kalkan, N.: Very high and ultrahigh frequency hyperchaotic oscillators with delay line. Chaos Solitons Fractals 17, 343 (2003) zbMATHCrossRefGoogle Scholar
  38. 38.
    Tamaševičius, A., Pyragine, T., Meskauskas, M.: Two scroll attractor in a delay dynamical system. Int. J. Bifurc. Chaos Appl. Sci. Eng. 17(10), 3455–3460 (2007) zbMATHCrossRefGoogle Scholar
  39. 39.
    Buscarino, A., Fortuna, L., Frasca, M., Sciuto, G.: Design of time-delay chaotic electronic circuits. IEEE Trans. Circuits Syst. I, Fundam. Theory Appl. 58, 1888–1896 (2011) MathSciNetCrossRefGoogle Scholar
  40. 40.
    Pham, V.-T., Fortuna, L., Frasca, M.: Implementation of chaotic circuits with a digital time-delay block. Nonlinear Dyn. 67, 345–355 (2012) CrossRefGoogle Scholar
  41. 41.
    Leonov, G.A., Kuznetsov, N.V.: Time-varying linearization and the Perron effects. Int. J. Bifurc. Chaos Appl. Sci. Eng. 17, 1079–1107 (2007) MathSciNetzbMATHCrossRefGoogle Scholar
  42. 42.
    Krasovskii, N.N.: Stability of Motion. Stanford University Press, Stanford (1963) zbMATHGoogle Scholar
  43. 43.
    Sedra, A.S., Smith, K.C.: Microelectronic Circuits. Oxford Univ. Press, London (2003) Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2012

Authors and Affiliations

  • Tanmoy Banerjee
    • 1
  • Debabrata Biswas
    • 1
  • B. C. Sarkar
    • 1
  1. 1.Department of PhysicsThe University of BurdwanBurdwanIndia

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