Optimize design of adaptive synchronization controllers and parameter observers in different hyperchaotic systems. (English) Zbl 1181.93032

Summary: Based on the Lyapunov stability and adaptive synchronization theory, optimization design of adaptive controllers and parameter observers with controllable gain coefficient are investigated in detail. The linear errors of corresponding variables and parameters are used to construct different appropriate positive Lyapunov functions \(V\) and the parameter observers and adaptive controllers are approached analytically by simplifying the differential inequality \(dV/dt \leqslant 0\). Particularly, an optional gain coefficient is selected in the parameter observers and positive Lyapunov function, which decides the transient period to identify the unknown parameters and reach synchronization. The scheme is used to study the synchronization of two non-identical hyperchaotic Rössler systems. The theoretical and numerical results confirm that the four unknown parameters in the drive system are estimated exactly and the two hyperchaotic systems reach complete synchronization when the controllers and parameter observers work on the driven system. To confirm the model independence of this scheme, an alternative hyperchaotic system is investigated, whereby the results confirm that the five unknown parameters are identified rapidly and exactly, and that the two hyperchaotic systems reach complete synchronization as well.


93B40 Computational methods in systems theory (MSC2010)
93A05 Axiomatic systems theory
34H10 Chaos control for problems involving ordinary differential equations
93C15 Control/observation systems governed by ordinary differential equations
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[1] Boccaletti, S.; Grebogi, C.; Lai, Y. C.; Mancini, H.; Maza, D., The control of chaos: theory and applications, Phys. Rep., 329, 103-197 (2000)
[2] Boccaletti, S.; Kurths, J.; Osipov, G.; Valladares, D. L.; Zhou, C. S., The synchronization of chaotic systems, Phys. Rep., 366, 1-101 (2002) · Zbl 0995.37022
[3] Perc, M., Visualizing the attraction of strange attractors, Eur. J. Phys., 26, 579-587 (2005)
[4] Wang, Q. Y.; Lu, Q. S.; Chen, G. R.; Guo, D. H., Chaos synchronization of coupled neurons with gap junction, Phys. Lett. A, 356, 17-25 (2006) · Zbl 1160.81304
[5] Perc, M.; Marh, M., Chaos in temporarily destabilized regular systems with the slow passage effect, Chaos, Solitons & Fractals, 27, 395-440 (2006) · Zbl 1149.92300
[6] Perc, M.; Marhl, M., Detecting and controlling unstable periodic orbits that are not part of a chaotic attractor, Phys. Rev. E, 70, 016204 (2004)
[7] Pecoral, L. M.; Carroll, T. L., Synchronization in chaotic systems, Phys. Rev Lett., 64, 821-824 (1990) · Zbl 0938.37019
[8] Ott, E.; Grebogi, C.; Yorke, J. A., Controlling chaos, Phys. Rev. Lett., 64, 1196-1199 (1990) · Zbl 0964.37501
[9] Maybhate, A.; Amritkar, R. E., Use of synchronization and adaptive control in parameter estimation from a time series, Phys. Rev. E, 59, 284-293 (1999)
[10] Wei, D. Q.; Luo, X. S., Adaptive robust control of chaotic oscillations in power system with excitation limits, Chinese Phys., 16, 11, 3244-3248 (2007)
[11] Wei, D. Q.; Luo, X. S.; Wang, B. H.; Fang, J. Q., Robust adaptive dynamic surface control of chaos in permanent magnet synchronous motor, Phys. Lett. A, 363, 71-77 (2007)
[12] Wei, D. Q.; Zhang, B., Finite time control for chaos in permanent magnet synchronous motor, Chinese Phys., 18, 1399-1405 (2009)
[13] d’Anjou, A.; Sarasola, C.; Torrealdea, F. J., Parameter-adaptive identical synchronization disclosing Lorenz chaotic masking, Phys. Rev. E, 63, 046213 (2001)
[14] Feki, M., An adaptive chaos synchronization scheme applied to secure communication, Chaos, Solitons & Fractals, 18, 141-148 (2003) · Zbl 1048.93508
[15] Yassen, M. T., Adaptive control and synchronization of a modified Chua’s system, Appl. Math. Comput., 135, 113-128 (2003) · Zbl 1038.34041
[16] Chen, S. H.; Lü, J. A., Synchronization of an uncertain unified chaotic system via adaptive control, Chaos, Solitons & Fractals, 14, 643-647 (2002) · Zbl 1005.93020
[17] Lu, J. Q.; Cao, J. D., Adaptive complete synchronization of two identical or different chaotic (hyperchaotic) systems with fully unknown parameters, Chaos, 15, 043901 (2005) · Zbl 1144.37378
[18] Fotsin, H. B.; Daafouz, J., Adaptive synchronization of uncertain chaotic colpitts oscillators based on parameter identification, Phys. Lett. A, 339, 304-315 (2005) · Zbl 1145.93313
[19] Cai, G. L.; Zhang, S.; Tian, L. X., Adaptive control and synchronization of an uncertain new hyperchaotic Lorenz system, Chinese Phys. Bull., 17, 2412-2419 (2008)
[20] Wang, Q. Y.; Lu, Q. S.; Chen, G. R., Synchronization transition by synaptic delay in coupled fast spiking neurons, Int. J. Bifurcat. Chaos, 4, 1189-1198 (2008) · Zbl 1147.34334
[21] Perc, M.; Marhl, M., Synchronization of regular and chaotic oscillations: the role of local divergence and the slow passage effect – a case study on calcium oscillations, Int. J. Bifurcat. Chaos, 14, 2735-2751 (2004) · Zbl 1065.92015
[22] Liu, C. X., A new hyperchaotic dynamical system, Chinese Phys., 16, 3279-3284 (2007)
[23] Wang, Z. L., Anti-synchronization in two non-identical hyperchaotic systems with known or unknown parameters, Commun. Non. Sci. Num. Sim., 14, 5, 2366-2372 (2009)
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