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.
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