The Fundamental Theorem of Dynamical Systems: all at once and all in the same place
Dynamical Systems
2025-08-15 v1
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.
Keywords
Cite
@article{arxiv.2508.10525,
title = {The Fundamental Theorem of Dynamical Systems: all at once and all in the same place},
author = {Andrew D. Lewis},
journal= {arXiv preprint arXiv:2508.10525},
year = {2025}
}