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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.

##### MSC:
 34C15 Nonlinear oscillations, coupled oscillators (ODE) 34C28 Complex behavior, chaotic systems (ODE) 34H05 ODE in connection with control problems 93C99 Control systems, guided systems 94C05 Analytic circuit theory 34C60 Qualitative investigation and simulation of models (ODE)
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