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Adaptive chaos synchronization in Chua’s systems with noisy parameters. (English) Zbl 1166.34029

The Chua’s system is modeled by the following differential equations

${\stackrel{˙}{x}}_{1}=p\left({x}_{2}-\frac{1}{7}\left(2{x}_{1}^{3}-{x}_{1}\right)\right),\phantom{\rule{1.em}{0ex}}{\stackrel{˙}{x}}_{2}={x}_{1}-{x}_{2}+{x}_{3},\phantom{\rule{1.em}{0ex}}{\stackrel{˙}{x}}_{3}=-q{x}_{2}+r{x}_{1}^{2},\phantom{\rule{2.em}{0ex}}\left(1\right)$

where $p$, $q$ and $r$ are real positive constants, and ${x}_{i}$, $i=1,\cdots ,3$, are state variables. The parameters $p,q,r$ are unknown and in additional, due to system uncontrollable variations, their actual values deviate randomly around their mean values.

The synchronization problem of two chaotic Chua systems whose coefficients are unknown and stochastically time varying is studied. Stochastic behaviour of the parameters is modeled by white Gaussian noise generated by a Wiener process. Using the Lyapounov stability theory, a Markov adaptive control law is designed for synchronizing the stochastic chaotic behaviour of the two systems in the sense of mean value convergence. Simulation results indicate that the proposed adaptive controller has a high performance in synchronization of chaotic Chua circuits in noisy environment.

##### MSC:
 34D05 Asymptotic stability of ODE 34F05 ODE with randomness 34C28 Complex behavior, chaotic systems (ODE) 34C15 Nonlinear oscillations, coupled oscillators (ODE) 34H05 ODE in connection with control problems 93C40 Adaptive control systems
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