2025/07/15 by István Balázs, Balázs, István, Ábel Garab +2
Physics and Astronomy · #34K25 #34K43 #37B35 #37C10 #37C70 #Dynamical Systems (math.DS) #FOS: Mathematics #Quantum chaos and dynamical systems
paper · pdf · doi:10.48550/arxiv.2507.11382
openalex publication_date 2025/07/15 · openalex created_date 2025/10/08 · openalex updated_date 2026/07/28
Understanding the structure of the global attractor is crucial in the field of dynamical systems, where Morse decompositions provide a powerful tool by partitioning the attractor into finitely many invariant Morse sets and gradient-like connecting orbits. Building on Mallet-Paret's pioneering use of discrete Lyapunov functions for constructing Morse decompositions in delay differential equations, similar approaches have been extended to various delay systems, also including state-dependent delays. In this paper, we develop a unified framework assuming the existence and some properties of a discrete Lyapunov function for a semi-dynamical system on an arbitrary metric space, and construct a Morse decomposition of the global attractor in this general setting. We demonstrate that our findings generalize previous results; moreover, we apply our theorem to a cyclic system of differential equations with threshold-type state-dependent delay.