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A linear feedback synchronization theorem for a class of chaotic systems. (English) Zbl 1032.34045

The authors consider the problem of synchronization for the following unidirectionally coupled system \[ x' = Ax+F(x),\quad y' = Ay+F(y) + K(x_n-y_n), \] where \(x=(x_1,\dots,x_n)^T\in \mathbb{R}^n\), \(y=(y_1,\dots,y_n)^T\in \mathbb{R}^n\), \(F\) has the special form \(F(x)=(f_1(x_1,\dots,x_n),f_2(x_2,\dots,x_n), \dots,f_n(x_n))^T\). The matrix \(A=\{a_{ij}\}\) is assumed to have some zero elements \(a_{ij}=0\) if \(j<i-1\). The main result of the paper gives conditions under which one can construct the feedback gain \(K\) to achieve a global synchronization in the sense that \(\lim_{t\to\infty}\|x(t)-y(t)\|=0\).

MSC:

34D05 Asymptotic properties of solutions to ordinary differential equations
34C28 Complex behavior and chaotic systems of ordinary differential equations
34H05 Control problems involving ordinary differential equations
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References:

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