We describe topological dynamics over a space by starting from a simple ODE emerging out of two coupled variables. We describe the dynamics of the evolution of points in space within the deterministic and stochastic frameworks. Historically dynamical systems were associated with celestial mechanics. The core philosophies of two kinds of dynamics emerging from Poincaré and Lyapunov are described. Smale’s contributions are highlighted. Markovian models are considered. Semi-group actions are a tool in this study.

中文翻译：

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动力系统：从经典力学和天文学到现代方法

我们通过从两个耦合变量中出现的简单 ODE 开始描述空间上的拓扑动力学。我们描述了确定性和随机框架内空间点演化的动态。历史上的动力系统与天体力学有关。描述了从庞加莱和李雅普诺夫出现的两种动力学的核心哲学。突出了 Smale 的贡献。考虑马尔可夫模型。半群体行动是本研究中的一个工具。