Abstract
When two different chaotic oscillators are coupled, generalized synchronization can occur. It may imply a very complicated relation between the states of drive and response systems. We propose a method that can be used to detect and characterize the generalized synchronization in modulated time-delayed systems. Using Krasovskii–Lyapunov theory, sufficient condition for generalized synchronization is derived. The proposed technique has been applied to synchronize prototype and Ikeda models by numerical simulation.
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References
Pecora, L.M., Carroll, T.L.: Synchronization in chaotic systems. Phys. Rev. Lett. 64, 821–824 (1990)
Pikovsky, A., Rosenblum, M., Kurths, J.: Synchronization: A Universal Concept in Nonlinear Sciences. Cambridge University Press, Cambridge (2001)
Pikovsky, A., Rosenblum, M., Kurths, J.: Phase synchronization of chaotic oscillations in terms of periodic orbits. Chaos 7(4), 680–687 (1997). Special issue on chaos synchronization
Chen, G., Dong, X.: From Chaos to Order. Methodologies, Perspectives and Applications. World Scientific, Singapore (1998)
Schuster, H.G. (ed.): Handbook of Chaos Control. Wiley, Weinheim (1999)
Fujisaka, H., Yamada, T.: Stability theory of synchronized motion in coupled-oscillator systems. Prog. Theor. Phys. 69, 32–47 (1983)
Rosenblum, M.G., Pikovsky, A.S., Kurths, J.: Phase synchronization of chaotic oscillators. Phys. Rev. Lett. 76, 1804–1807 (1996)
Yalcinkaya, T., Lai, Y.C.: Phase characterization of chaos. Phys. Rev. Lett. 79, 3885–3888 (1997)
Senthilkumar, D.V., Lakshmanan, M., Kurths, J.: Phase synchronization in time-delay systems. Phys. Rev. E 74, 035205(R) (2006)
Zhan, M., Wei, G.W., Lai, C.H.: Transition from intermittency to periodicity in lag synchronization in coupled Rössler oscillators. Phys. Rev. E 65, 036202 (2002)
Voss, H.U.: Dynamic long-term anticipation of chaotic states. Phys. Rev. Lett. 87, 014102 (2001)
Rulkov, N.F., Suschik, 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)
Kocarev, L., Parlitz, U.: Generalized synchronization, predictability, and equivalence of unidirectionally coupled dynamical systems. Phys. Rev. Lett. 76, 1816–1819 (1996)
Banerjee, S., Ghosh, D., Roy Chowdhury, A.: Multiplexing synchronization and its applications in cryptography. Phys. Scr. 78, 015010 (2008)
Abarbanel, H.D.I., Rulkov, N.F., Suschik, M.M.: Generalized synchronization of chaos: The auxiliary system approach. Phys. Rev. E 53, 4528–4535 (1996)
Pecora, L.M., Carroll, T.L., Heagy, J.F.: Statistics for mathematical properties of maps between time series embeddings. Phys. Rev. E 52, 3420–3439 (1995)
Juan, M., Xingyuan, W.: Generalized synchronization via nonlinear control. Chaos 18, 023108 (2008)
Hramov, E., Koronovskii, A.A.: Generalized synchronization: A modified system approach. Phys. Rev. E 71, 067201 (2005)
Yang, T., Chua, L.O.: Generalized synchronization of chaos via linear transforms. Int. J. Bifurc. Chaos Appl. Sci. Eng. 9, 215–219 (1999)
Kale, J.K., Lunel, S.M.V.: Introduction to Functional Differential Equations. Springer, New York (1993)
Arecchi, F.T., Meucci, R., Allaria, E., Di Garbo, A., Tsimring, L.S.: Delayed self-synchronization in homoclinic chaos. Phys. Rev. E 65, 046237 (2002)
Hegger, R., Bunner, M.J., Kantz, H., Giaquinta, A.: Identifying and modeling delay feedback systems. Phys. Rev. Lett. 81, 558–561 (1998)
Bunner, M.J., Meyer, Th., Kittel, A., Parisi, J.: Recovery of the time-evolution equation of time-delay systems from time series. Phys. Rev. E 56, 5083–5089 (1997)
Ponomarenko, V.I., Prokhorov, M.D.: Extracting information masked by the chaotic signal of a time-delay system. Phys. Rev. E 66, 026215 (2002)
Zhou, C., Lai, C.-H.: Extracting messages masked by chaotic signals of time-delay systems. Phys. Rev. E 60, 320–323 (1999)
Short, K.M., Parker, A.T.: Unmasking a hyperchaotic communication scheme. Phys. Rev. E 58, 1159–1162 (1998)
Ghosh, D., Banerjee, S.: Adaptive scheme for synchronization-based multiparameter estimation form a single chaotic time series and its applications. Phys. Rev. E 78, 056211 (2008)
Ghosh, D.: Nonlinear observer-based synchronization scheme for multiparameter estimation. Europhys. Lett. 84, 40012 (2008)
Grassi, G., Mascolo, S.: Nonlinear observer design to synchronize hyperchaotic systems via a scalar signal. IEEE Trans. Circ. Syst. 1: Fundam. Theory Appl. 44, 1011 (1997)
Shahverdiev, E.M., Shore, K.A.: Generalized synchronization in time-delayed systems. Phys. Rev. E 71, 016201 (2005)
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)
Tamasevicius, A., Cenys, A., Mykolaitis, G., Namajunas, A., Lindberg, E.: Hyperchaotic oscillator with gyrators. IEEE Electron. Lett. 33, 542–543 (1997)
Ikeda, K., Daido, H., Akimoto, O.: Optical turbulence: Chaotic behavior of transmitted light from a ring cavity. Phys. Rev. Lett. 45, 709–712 (1980)
Ikeda, K., Kondo, K., Akimoto, O.: Successive higher-harmonic bifurcations in systems with delayed feedback. Phys. Rev. Lett. 49, 1467–1470 (1982)
Goedgebuer, J.P., Larger, L., Porte, H.: Chaos in wavelength with a feedback tunable laser diode. Phys. Rev. E 57, 2795–2798 (1998)
Mackey, M.C., Glass, L.: Oscillation and chaos in physiological control systems. Science 197, 287–289 (1977)
Ghosh, D., Banerjee, S., Roy Chowdhury, A.: Synchronization between variable time-delayed systems and cryptography. Europhys. Lett. 80, 30006 (2007)
Banerjee, S., Ghosh, D., Ray, A., Roy Chowdhury, A.: Synchronization between two different time-delayed systems and image encryption. Europhys. Lett. 81, 20006 (2008)
Ghosh, D.: Generalized projective synchronization in time-delayed systems: Nonlinear observer approach. Chaos 19, 013102 (2009)
Krasovskii, N.N.: Stability of Motion. Stanford University Press, Stanford (1963)
Pyragas, K.: Synchronization of coupled time-delay systems: Analytical estimations. Phys. Rev. E 58, 3067–3071 (1998)
Ucar, A.: On the chaotic behaviour of a prototype delayed dynamical system. Chaos Solitons Fractals 16, 187–194 (2003)
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Ghosh, D. Nonlinear active observer-based generalized synchronization in time-delayed systems. Nonlinear Dyn 59, 289–296 (2010). https://doi.org/10.1007/s11071-009-9538-4
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DOI: https://doi.org/10.1007/s11071-009-9538-4