We analyze the chain recurrent set of skew product semiflows obtained from nonautonomous differential equations—ordinary differential equations or semilinear parabolic differential equations. For many gradient-like dynamical systems, Morse–Smale dynamical systems e.g., the chain recurrent set contains only isolated equilibria. The structure in the asymptotically autonomous setting is richer but still close to the structure of a Morse–Smale dynamical system. The main tool used in this paper is a nonautonomous flavour of Conley index theory developed by the author. We will see that for a class of good equations, the Conley index can be understood in terms of equilibria (in a generalized meaning) and their connections. This allows us to find specific solutions of asymptotically autonomous equations and generalizes properties of Morse–Smale dynamical systems to the asymptotically autonomous setting.

中文翻译：

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渐近自治微分方程的平衡类解

我们分析了从非自治微分方程（普通微分方程或半线性抛物线微分方程）获得的偏积半流的链递归集。对于许多类似梯度的动力系统，摩尔斯-斯马德动力系统（例如，链递归集）仅包含孤立的平衡点。渐近自治环境中的结构较丰富，但仍接近摩尔斯-斯马德动力学系统的结构。本文使用的主要工具是作者开发的非自治的Conley指数理论。我们将看到，对于一类好的方程，可以用均衡（广义上）及其联系来理解Conley指数。