Morse decomposition for semi-dynamical systems with an application to systems of state-dependent delay differential equations
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866916844742377472 |
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| author | Balázs, István Garab, Ábel Rauscher, Teresa |
| author_facet | Balázs, István Garab, Ábel Rauscher, Teresa |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_11382 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Morse decomposition for semi-dynamical systems with an application to systems of state-dependent delay differential equations Balázs, István Garab, Ábel Rauscher, Teresa Dynamical Systems 37B35, 37C70, 37C10, 34K43, 34K25 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. |
| title | Morse decomposition for semi-dynamical systems with an application to systems of state-dependent delay differential equations |
| topic | Dynamical Systems 37B35, 37C70, 37C10, 34K43, 34K25 |
| url | https://arxiv.org/abs/2507.11382 |