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Generalized projective synchronization for the chaotic Lorenz system and the chaotic Chen system. (English) Zbl 1118.34050

Two systems are said to have the “projective synchronization” if their state variables \(x\in \mathbb{R}^n\) and \(y \in \mathbb{R}^n\) satisfy asymptotically \(\| x(t)-\alpha y(t)\| \to 0\) for \(t\to \infty\) and all initial conditions with some scalar \(\alpha\). The authors consider the following control setup in order to achieve the projective synchronization: \[ x'=f(x),\quad y'=f(y) + u, \] where the control \(u(x,y)\) is chosen in such a way that the error system admits the linear form \((\alpha y-x)'=M(\alpha y-x)\) with a stable matrix \(M\).

MSC:

34H05 Control problems involving ordinary differential equations
34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations
34D35 Stability of manifolds of solutions to ordinary differential equations
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