Morse decomposition for semi-dynamical systems with an application to systems of state-dependent delay differential equations

István Balázs; Ábel Garab; Teresa Rauscher

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arXiv · 2507.11382

Morse decomposition for semi-dynamical systems with an application to systems of state-dependent delay differential equations

István Balázs·Ábel Garab·Teresa Rauscher

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Abstract

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.

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BibTeX RIS

István Balázs, Ábel Garab, Teresa Rauscher. 2025-07-15. Morse decomposition for semi-dynamical systems with an application to systems of state-dependent delay differential equations. https://arxiv.org/abs/2507.11382

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