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Adaptive control and synchronization of a modified Chua’s circuit system. (English) Zbl 1038.34041
The problem of chaos control is considered for the modified Chua circuit $$x'=p(y-\frac{1}{7}(2x^3 -x )), \qquad y' = x-y+z, \qquad z=-qy.$$ Conditions are given for the asymptotic stabilization of a fixed-point in the presence of the feedback control term $-k(x-\bar x)$, which is introduced in the equation for the $x$ variable. $\bar x$ is the $x$-component of the fixed-point under consideration. This point is shown to be unstable in the absence of a controller. Then, the author derives conditions for the stabilization of the fixed-point in the presence of adaptive controller $-g(x-\bar x)$, where $g$ is updated according to the following algorithm $g'=\gamma (x-\bar x)^2$. Finally, adaptive synchronization of two unidirectionally coupled identical Chua systems is considered.

34C15Nonlinear oscillations, coupled oscillators (ODE)
34C28Complex behavior, chaotic systems (ODE)
34H05ODE in connection with control problems
93C99Control systems, guided systems
94C05Analytic circuit theory
34C60Qualitative investigation and simulation of models (ODE)
Full Text: DOI
[1] Ott, E.; Grebogi, C.; Yorke, J. A.: Phys. rev. Lett.. 64, No. 11, 1196-1199 (1990)
[2] Rajasekar, S.; Murali, K.; Lakshmanan, M.: Chaos, soliton and fractals. 8, No. 9, 1545-1558 (1997)
[3] Ramesh, M.; Narayanan, S.: Chaos, soliton and fractals. 10, No. 9, 1473-1489 (1999) · Zbl 0983.37040
[4] Chen, G.; Dong, X.: IEEE trans. Circuits and systems. 40, No. 9, 591-601 (1993) · Zbl 0800.93758
[5] Chen, G.; Dong, X.: J. circuits and systems comput.. 3, No. 1, 139-149 (1993)
[6] Chen, G.: IEEE proc. Of amer. Contr. conf., San Francisco, CA. 2413-2414 (1993)
[7] Chen, G.: Chaos, soliton and fractals. 8, No. 9, 1461-1470 (1997)
[8] Hartley, T. T.; Mossayebi, F.: J. circuits and systems comput.. 3, 173-194 (1993)
[9] Saito, T.; Mitsubori, K.: IEEE trans. Circuits and systems. 42, 168-172 (1995)
[10] Hwang, C. C.; Chow, H. Y.; Wang, Y. K.: Physica D. 92, 95-100 (1996)
[11] Hwang, C. C.; Hsieh, J. Y.; Lin, R. S.: Chaos, soliton and fractals. 8, No. 9, 1507-1515 (1997)
[12] Pyragas, K.: Phys. lett. A. 170, 421-428 (1992)
[13] Hegazi, A.; Agiza, H. N.; El-Dessoky, M. M.: Chaos, soliton and fractals. 12, 631-658 (2001) · Zbl 1016.37050
[14] Pecora, L. M.; Carrol, T. L.: Phys. rev. Lett.. 64, No. 8, 821-824 (1990)
[15] Agiza, H. N.; Yassen, M. T.: Phys. lett. A. 278, 191-197 (2001)
[16] Liao, T. L.; Lin, S. H.: J. franklin inst.. 336, 925-937 (1999)
[17] Bai, E. W.; Lonngren, K. E.: Chaos, soliton and fractals. 11, 1041-1044 (2000)
[18] Di Bernardo, M.: Phys. lett. A. 214, 139-144 (1996) · Zbl 0972.93509
[19] Madan, R. N.: Chua’s circuit, A paradigm for chaos. (1993) · Zbl 0861.58026
[20] Sparrow, C.: The Lorenz equations bifurcation, chaos and strange attractors. (1982) · Zbl 0504.58001