×

Output synchronization of chaotic systems under nonvanishing perturbations. (English) Zbl 1141.93391

Summary: An analysis for chaos synchronization under nonvanishing perturbations is presented. In particular, we use model-matching approach from nonlinear control theory for output synchronization of identical and nonidentical chaotic systems under nonvanishing perturbations in a master-slave configuration. We show that the proposed approach is indeed suitable to synchronize a class of perturbed slaves with a chaotic master system; that is the synchronization error trajectories remain bounded if the perturbations satisfy some conditions. In order to illustrate this robustness synchronization property, we present two cases of study: (i) for identical systems, a pair of coupled Rössler systems, the first like a master and the other like a perturbed slave, and (ii) for nonidentical systems, a Chua’s circuit driving a Rössler/slave system with a perturbed control law, in both cases a quantitative analysis on the perturbation is included.

MSC:

93D05 Lyapunov and other classical stabilities (Lagrange, Poisson, \(L^p, l^p\), etc.) in control theory
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
37N35 Dynamical systems in control
93C73 Perturbations in control/observation systems
93C10 Nonlinear systems in control theory
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Pecora, L. M.; Carroll, T. L., Synchronization in chaotic systems, Phys Rev Lett, 64, 8, 821-824 (1990) · Zbl 0938.37019
[2] Special issue on control and synchronization of chaos, Int J Bifurcat Chaos, 10, 3-4 (2000)
[3] Pikovsky, A.; Rosenblum, M.; Kurths, J., Synhcronization: a universal concept in nonlinear sciences (2001), Cambridge University Press: Cambridge University Press Cambridge
[4] Synchronization through extended Kalman filtering, (Nijmeijer, H.; Fossen, T. I., New trends in nonlinear observer design. New trends in nonlinear observer design, Lecture notes in control and information sciences, vol. 244 (1999), Springer: Springer London), 469-490 · Zbl 0927.93053
[5] Sira-Ramírez, H.; Cruz-Hernández, C., Synchronization of chaotic systems: a generalized Hamiltonian systems approach, Int J Bifurcat Chaos, 11, 5, 1381-1395 (2001), And in: Proceedings of the American Control Conference, Chicago, USA, 2000. pp. 769-773. · Zbl 1206.37053
[6] Aguilar, A. Y.; Cruz-Hernández, C., Synchronization of two hyperchaotic Rössler systems: model-matching approach, WSEAS Trans Syst, 1, 2, 198-203 (2002)
[7] López-Mancilla, D.; Cruz-Hernández, C., Output synchronization of chaotic systems: model-matching approach with application to secure communication, Nonlinear Dyn Syst Theory, 5, 2, 141-156 (2005) · Zbl 1097.37056
[8] Feldmann, U.; Hasler, M.; Schwarz, W., Communication by chaotic signals: the inverse system approach, Int J Circ Theory Appl, 24, 5, 551-579 (1996) · Zbl 0902.94005
[9] Nijmeijer, H.; Mareels, I. M.Y., An observer looks at synchronization, IEEE Trans Circ Syst I, 44, 10, 882-890 (1997)
[10] Kocarev, L.; Halle, K. S.; Eckert, K.; Chua, L. O., Experimental demonstration of secure communications via chaotic synchronization, Int J Bifurcat Chaos, 2, 3, 709-713 (1992) · Zbl 0875.94134
[11] Cuomo, K. M.; Oppenheim, A. V.; Strogratz, S. H., Synchronization of Lorenz-based chaotic circuits with applications to communications, IEEE Trans Circ Syst II, 40, 10, 626-633 (1993)
[12] Wu, C. W.; Chua, L. O., A simple way to synchronize chaotic systems with applications to secure communication systems, Int J Bifurcat Chaos, 3, 6, 1619-1627 (1993) · Zbl 0884.94004
[13] Posadas-Castillo, C.; Cruz-Hernández, C.; Núñez, R., Experimental realization of binary signals transmission based on synchronized Lorenz circuits, J Appl Res Technol, 2, 2, 127-137 (2004)
[14] Cruz-Hernández, C.; López-Mancilla, D.; García, V.; Serrano, H.; Núñez, R., Experimental realization of binary signal transmission using chaos, J Circ Syst Comput, 14, 3, 453-568 (2005)
[15] Cruz-Hernández, C., Synchronization of time-delay Chua’s oscillator with application to secure communication, Nonlinear Dyn Syst Theory, 4, 1, 1-13 (2004) · Zbl 1058.93027
[16] Cruz-Hernández C, Romero-Haros N. Communicating via synchronized time-delay Chua’s circuits. Commun Nonlinear Sci Numer Simul in press, doi:10.1016/j.cnsns.2006.06.010; Cruz-Hernández C, Romero-Haros N. Communicating via synchronized time-delay Chua’s circuits. Commun Nonlinear Sci Numer Simul in press, doi:10.1016/j.cnsns.2006.06.010 · Zbl 1121.94028
[17] Mormann, F.; Lehnertz, K.; David, P.; Elger, C. E., Mean phase coherence as a measure for phase synchronization and its application to the EEG of epilepsy patients, Physica D, 144, 358-369 (2000) · Zbl 0962.92020
[18] Rabinovich, M. I.; Varona, P.; Abarbanel, H. D.I., Nonlinear cooperative dynamics of living neurons, Int J Bifurcat Chaos, 10, 5, 913-933 (2000) · Zbl 1090.92506
[19] Tass, P.; Rosenblum, M. G.; Weule, J.; Kurths, J.; Pikovsky, A.; Volkmmann, J., Detection of \(n\):\(m\) phase locking from noisy data: application to magnetoencephalography, Phys Rev Lett, 81, 15, 3291-3294 (1998)
[20] Lotric, M. B.; Stefanovska, A., Synchronization and modulation in the human cardiorespiratory system, Physica A, 283, 3-4, 451-561 (2000)
[21] Winfree, A. T., Biological rhythms and the behavior of populations of coupled oscillators, J Theor Biol, 16, 15-42 (1967)
[22] Castro R, Di Benedetto MD. Asymptotic nonlinear model matching. In: Proceedings of the 29th IEEE conference on decision and control, Hawaii, USA, 1990. p. 3400-3403.; Castro R, Di Benedetto MD. Asymptotic nonlinear model matching. In: Proceedings of the 29th IEEE conference on decision and control, Hawaii, USA, 1990. p. 3400-3403.
[23] Di Benedetto, M. D.; Grizzle, J. W., Asymptotic model matching for nonlinear systems, IEEE Trans Automat Control, 39, 8, 1539-1550 (1994) · Zbl 0815.93006
[24] Isidori, A., Nonlinear control systems (1995), Springer-Verlag: Springer-Verlag London · Zbl 0569.93034
[25] Cruz-Hernández, C.; Alvarez, J.; Castro, R., Stability of discrete nonlinear systems under nonvanishing perturbations: application to a nonlinear model-matching problem, IMA J Math Control Inf, 16, 23-41 (1999) · Zbl 0919.93060
[26] Cruz-Hernández, C.; Alvarez, J.; Castro, R., Stability robustness of linearizing controllers with state estimation for discrete-time nonlinear systems, IMA J Math Control Inf, 18, 479-489 (2001) · Zbl 0999.93066
[27] Kotta, U., Inversion method in the discrete-time control systems synthesis problems. Inversion method in the discrete-time control systems synthesis problems, Lecture notes in control and information sciences (1995), Springer-Verlag · Zbl 0822.93001
[28] López-Mancilla, D.; Cruz-Hernández, C., An analysis of robustness on the synchronization of chaotic systems under nonvanishing perturbation using sliding modes, WSEAS Trans Math, 3, 2, 364-369 (2004)
[29] Khalil, H. K., Nonlinear systems (1996), Prentice Hall · Zbl 0626.34052
[30] Rössler, O. E., An equation for continuous chaos, Phys Lett A, 57, 397-398 (1976) · Zbl 1371.37062
[31] Madan RN, Guest editor. Chua’s circuit: a paradigm for chaos. Singapore: World Scientific; 1993.; Madan RN, Guest editor. Chua’s circuit: a paradigm for chaos. Singapore: World Scientific; 1993.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.