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Abstract methods for synchronization and applications. (English) Zbl 0890.34052

Consider the abstract differential system \[ dx/dt= Ax+ G(x) \tag{*} \] in a real Banach space \(X\) where \(A\) is an infinitesimal generator of a \(C_0\)-semigroup, and \(G:X \to X\) is continuously differentiable and Lipschitzian on bounded sets. Assuming the existence of a Lyapunov-like function the author derives conditions to obtain uniform estimates for the global attractor of (*). In case of coupled systems \[ dx/dt =Ax+ G(x,\lambda_1) -K(x-y),\;dy/dt= Ay+ G(y,\lambda_2) -K(y-x) \tag{**} \] the same approach is used to establish synchronization in the sense that the solution components \(\overline x(t,x_1,y_1)\) and \(\overline y(t,x_1,y_1)\) satisfy \[ \bigl |\overline x(t,x_1,y_1) -\overline y(t,x_1, y_1) \bigr|\leq m_1 e^{-\alpha (t-t_0)} |x_1- y_1|+ m_2\gamma (\lambda_1-\lambda_2) \] for \(t\geq t_0\) where \(\alpha, m_1,m_2\) are positive constants, \(\gamma: \mathbb{R}\to \mathbb{R}_+\) is a continuous function obeying \[ \gamma (0)=0,\;\bigl|G(x,\lambda_2)-G(x,\lambda_1) \bigr |\leq\gamma (\lambda_2- \lambda_1). \] Some examples including coupled lasers and Lorenz systems are discussed.

MSC:

34G20 Nonlinear differential equations in abstract spaces
34D45 Attractors of solutions to ordinary differential equations
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