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

2025/08/14 by Lewis, Andrew D.
#34A12 #46E10 #46T05 #Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.2508.10525

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.

Related

• Dynamical Systems: Discrete, Continuous and Hybrid
• Notes on Chain Recurrence and Lyapunonv Functions
• Attractors and Attracting Neighborhoods for Multiflows
• Algebraic and Dynamic Lyapunov Equations on Time Scales
• Attractors as a bridge from topological properties to long-term behavior in dynamical systems