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Asymptotically autonomous semiflows: Chain recurrence and Lyapunov functions. (English) Zbl 0829.34037
A nonautonomous system \(x' = f(t,x)\), \(x \in \mathbb{R}^n\), is called asymptotically autonomous with limit equation \(y' = g(y)\), \(y \in \mathbb{R}^n\), if \(f(t,x)\) uniformly converges to \(g(x)\) on each compact as \(t \to \infty\). Let \(\Phi (t,s,x_0) = x(t)\) be the solution of the first system determined by \(x(s) = x_0\), i.e., \(\Phi\) is the asymptotically autonomous semiflow. Let \(\Theta (t,x_0) = y(t)\) be the solution of the second system determined by \(y(0) = x_0\); the autonomous semiflow. The authors investigate the \(\omega\)-limit set \(\omega (t_0, x_0)\) of a bounded solution \(x(t)\), \(x(t_0) = x_0\), consisting of all limits \(y = \lim x (t_j)\) as \(t_j \to \infty \). First of all, this set is chain recurrent in the following sense: to every \(y \in \omega (t_0, x_0)\), \(\varepsilon > 0\), \(t > 0\) there exist sequences \(x = x_1\), \(x_2, \ldots, x_n, x_{n + 1} = x \in \omega (t_0, x_0)\) and \(t_1, \ldots, t_{n + 1} > t\) such that \(\text{dist} (\Theta (t_i, x_i), x_{i + 1}) < \varepsilon\). Conversely, every nonempty, compact, connected, and chain recurrent set is the \(\omega\)-limit set of a solution of a system \(x' = g(y) + \psi(t)\) for appropriate choice of smooth function \(\psi (t)\). Sufficient conditions are given for an \(\omega\)-limit set to be contained in a level set of a Lyapunov function.
Reviewer: J.Chrastina (Brno)

MSC:
35B40 Asymptotic behavior of solutions to PDEs
37C80 Symmetries, equivariant dynamical systems (MSC2010)
34D20 Stability of solutions to ordinary differential equations
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