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Adaptive backstepping synchronization of uncertain chaotic system. (English) Zbl 1062.34053
The paper is devoted to the synchronization of systems, which can be represented in so-called strict-feedback form, i.e., $$\dot x_1=f_1(x_1,x_2)$$, $$\dot x_2 = f_2 (x_1,x_2,x_3)$$, $$\dots$$, $$\dot x_{n-1}=f_{n-1}(x_1,\dots,x_n)$$, $$\dot x_{n}=f_{n}(x_1,\dots,x_n)+g_1(t)$$, where $$f_1$$ is a linear function. The response system is coupled via the last $$n$$th component $$\dot {\bar x}_{n}=f_{n}(\bar x_1,\dots,\bar x_n)+g_2(t) +u$$.
The authors suggest an adaptive design of the coupling term $$u$$, including a law for the parameter adaptation such that the systems are synchronized, that is, $$| x(t)-\bar x(t)| \to \infty$$ as $$t\to \infty$$ for all initial conditions $$x(0)$$ and $$\bar x(0)$$.

##### MSC:
 34D05 Asymptotic properties of solutions to ordinary differential equations 34D23 Global stability of solutions to ordinary differential equations 34C28 Complex behavior and chaotic systems of ordinary differential equations