The Fundamental Theorem of Dynamical Systems: all at once and all in the same place

TL;DR

This paper provides a comprehensive overview of the Fundamental Theorem of Dynamical Systems, unifying continuous and discrete frameworks, and introduces a complete Lyapunov function for continuous-time flows and semiflows.

Contribution

It offers a unified approach to topological dynamics and introduces the first complete Lyapunov function for continuous-time flows and semiflows.

Findings

Unified framework for continuous and discrete dynamical systems.

First complete Lyapunov function for continuous-time flows.

Clarifies the relationship between attractors, repellers, and the chain recurrent set.

Abstract

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.