Abstract
In this paper, we investigate the projective synchronization between two different time-delayed chaotic systems. A suitable controller is chosen using the active control approach. We relax some limitations of previous work, where projective synchronization of different chaotic systems can be achieved only in finite dimensional chaotic systems, so we can achieve projective synchronization of different chaotic systems in infinite dimensional chaotic systems. Based on the Lyapunov stability theory, we suggest a generic method to achieve the projective synchronization between two different time-delayed chaotic systems. The validity of the proposed method is demonstrated and verified by observing the projective synchronization between two well-known time-delayed chaotic systems; the Ikeda system and Mackey–Glass system. Numerical simulations fully support the analytical approach.
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Pecora, L.M., Carroll, T.C.: Synchronization in chaotic systems. Phys. Rev. Lett. 64, 821–824 (1990)
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)
Rosenblum, M.G., Pikovsky, A.S., Kurths, J.: Phase synchronization of chaotic oscillators. Phys. Rev. Lett. 76, 1804–1807 (1996)
Mainieri, R., Rehacek, J.: Projective synchronization in three-dimensional chaotic systems. Phys. Rev. Lett. 82, 3042–3045 (1999)
Cao, L.Y., Lai, Y.C.: Antiphase synchronism in chaotic systems. Phys. Rev. E 58, 382–386 (1998)
Chee, C.Y., Xu, D.: Secure digital communication using controlled projective synchronisation of chaos. Chaos Solitons Fractals 23, 1063–1070 (2005)
Xu, D.: Control of projective synchronization in chaotic systems. Phys. Rev. E 63, 27201–27204 (2001)
Jia, Q.: Projective synchronization of a new hyperchaotic Lorenz system. Phys. Lett. A 370, 40–45 (2007)
Wen, G., Xu, D.: Nonlinear observer control for full-state projective synchronization in chaotic continuous-time systems. Chaos Solitons Fractals 26, 71–77 (2005)
Feng, C.F., Zhang, Y., Wang, Y.-H.: Projective synchronization in time-delayed chaotic systems. Chin. Phys. Lett. 23, 1418–1421 (2006)
Cao, J., Ho, D.W.C., Yang, Y.: Projective synchronization of a class of delayed chaotic systems via impulsive control. Phys. Lett. A 373, 3128–3133 (2009)
Ghosh, D.: Generalized projective synchronization in time-delayed systems: Nonlinear observer approach. Chaos 19, 013102 (2009)
Hu, M., Yang, Y., Xu, Z., Zhang, R., Guo, L.: Projective synchronization in drive-response dynamical networks. Physica A 381, 457–466 (2007)
Feng, C.F., Xu, X.-J., Wang, S.-J., Wang, Y.-H.: Projective-anticipating, projective, and projective-lag synchronization of time-delayed chaotic systems on random networks. Chaos 18, 023117-1-6 (2008)
Li, G.: Generalized projective synchronization between Lorenz system and Chen’s system. Chaos Solitons Fractals 32, 1454–1458 (2007)
Li, G., Zhou, S., Yang, K.: Generalized projective synchronization between two different chaotic systems using active backstepping control. Phys. Lett. A 355, 326–330 (2006)
Li, R., Xu, W., Li, S.: Adaptive generalized projective synchronization in different chaotic systems based on parameter identification. Phys. Lett. A 367, 199–206 (2007)
Traub, R.D., Miles, R., Wong, R.K.S.: Model of the origin of rhythmic population oscillations in the hippocampal slice. Science 243, 1319–1325 (1989)
Foss, J., Longtin, A., Mansour, B., Milton, J.: Multistability and delayed recurrent loops. Phys. Rev. Lett. 76, 708–711 (1996)
Pyragas, K.: Synchronization of coupled time-delay systems: Analytical estimations. Phys. Rev. E 58, 3067–3071 (1998)
Pyragas, K.: Transmission of signals via synchronization of chaotic time-delay systems. Int. J. Bifurc. Chaos 8, 1839–1842 (1998)
Masoller, C.: Spatiotemporal dynamics in the coherence collapsed regime of semiconductor lasers with optical feedback. Chaos 7, 455–462 (1997)
Bai, E.W., Lonngsen, K.E.: Sequential synchronization of two Lorenz systems using active control. Chaos Solitons Fractals 11, 1041–1044 (2000)
Bai, E.W., Lonngsen, K.E.: Synchronization of two Lorenz systems using active control. Chaos Solitons Fractals 8, 51–58 (1997)
Ho, M.C., Hung, Y.C., Chou, C.H.: Phase and anti-phase synchronization of two chaotic systems by using active control. Phys. Lett. A 296, 43–48 (2002)
Agiza, H.N., Yassen, M.T.: Synchronization of Rossler and Chen dynamical systems using active control. Phys. Lett. A 278, 191–197 (2001)
He, R., Vaiya, P.G.: Analysis and synthesis of synchronous periodic and chaotic systems. Phys. Rev. A 46, 7387–7392 (1992)
Ikeda, K., Kondo, K., Akimoto, O.: Successive higher-harmonic bifurcations in systems with delayed feedback. Phys. Rev. Lett. 49, 1467–1470 (1982)
Voss, H.U.: Dynamic long-term anticipation of chaotic states. Phys. Rev. Lett. 87, 014102-1-4 (2001)
Masoller, C.: Anticipation in the synchronization of chaotic semiconductor lasers with optical feedback. Phys. Rev. Lett. 86, 2782–2785 (2001)
Masoller, C., Zanette, D.H.: Anticipated synchronization in coupled chaotic maps with delays. Physica A 300, 359–366 (2001)
Shahverdiev, E.M.: Synchronization in systems with multiple time delays. Phys. Rev. E 70, 067202-1-4 (2004)
Namajūnas, A., Pyragas, K., Tamaševičius, A.: An electronic analog of the Mackey–Glass system. Phys. Lett. A 201, 42–46 (1995)
Kittel, A., Parisi, J., Pyragas, K.: Generalized synchronization of chaos in electronic circuit experiments. Physica D 112, 459–471 (1998)
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Feng, CF. Projective synchronization between two different time-delayed chaotic systems using active control approach. Nonlinear Dyn 62, 453–459 (2010). https://doi.org/10.1007/s11071-010-9733-3
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DOI: https://doi.org/10.1007/s11071-010-9733-3