Work

Morse Decomposition for Semi-Dynamical Systems with an Application to Systems of State-Dependent Delay Differential Equations

Year: 2025

Type: article

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.

Source: Journal of Dynamics and Differential Equations

DOI: 10.1007/s10884-025-10464-0

Authors: István Balázs, Ábel Garab, Teresa Rauscher

Institutions: University of Szeged, University of Klagenfurt

Language: English

Topic: Mathematical Dynamics and Fractals

Subfield: Mathematical Physics

Field: Mathematics

Domain: Physical Sciences

Open Access status: hybrid

Estimated APC paid: $3,090

Funders: Magyar Tudományos Akadémia, Szegedi Tudományegyetem, National Research, Development and Innovation Office