Morse Decomposition for Semi-Dynamical Systems with an Application to Systems of State-Dependent Delay Differential Equations
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...
Elmentve itt :
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| Dokumentumtípus: | Cikk |
| Megjelent: |
2025
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| Sorozat: | JOURNAL OF DYNAMICS AND DIFFERENTIAL EQUATIONS
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| Tárgyszavak: | |
| doi: | 10.1007/s10884-025-10464-0 |
| mtmt: | 36423634 |
| Online Access: | http://publicatio.bibl.u-szeged.hu/38233 |
| Tartalmi kivonat: | 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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| ISSN: | 1040-7294 |