Original manuscript: 2025/08/13

The so-called Fundamental Theorem of Dynamical Systems—which (1) relates attractors and repellers to the chain recurrent set and (2) gives the existence of a complete Lyapunov function—can be seen as a means of separating out ``recurrent'' and ``transient'' dynamics. An overview of this theorem is given in its various guises, continuous-time/discrete-time and flows/semiflows. As part of this overview, a unified approach is developed for working simultaneously with both the continuous-time and discrete-time frameworks for topological dynamics. Additionally, a complete Lyapunov function is provided for the first time for continuous-time flows and semiflows.

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Last Updated: Wed Aug 13 12:36:58 2025