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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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