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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2508.10525 |
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| _version_ | 1866918125058916352 |
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| author | Lewis, Andrew D. |
| author_facet | Lewis, Andrew D. |
| contents | The so-called Fundamental Theorem of Dynamical Systems -- which(1) relates attractors and repellers to the chain recurrent set and (2) gives the existence of a complete Lyapunov function -- can be seen as a means of separating out ``recurrent'' and ``transient'' dynamics. An overview of this theorem is given in its various guises, continuous-time/discrete-time and flows/semiflows. As part of this overview, a unified approach is developed for working simultaneously with both the continuous-time and discrete-time frameworks for topological dynamics. Additionally, a complete Lyapunov function is provided for the first time for continuous-time flows and semiflows. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_10525 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Fundamental Theorem of Dynamical Systems: all at once and all in the same place Lewis, Andrew D. Dynamical Systems 34A12, 46E10, 46T05 The so-called Fundamental Theorem of Dynamical Systems -- which(1) relates attractors and repellers to the chain recurrent set and (2) gives the existence of a complete Lyapunov function -- can be seen as a means of separating out ``recurrent'' and ``transient'' dynamics. An overview of this theorem is given in its various guises, continuous-time/discrete-time and flows/semiflows. As part of this overview, a unified approach is developed for working simultaneously with both the continuous-time and discrete-time frameworks for topological dynamics. Additionally, a complete Lyapunov function is provided for the first time for continuous-time flows and semiflows. |
| title | The Fundamental Theorem of Dynamical Systems: all at once and all in the same place |
| topic | Dynamical Systems 34A12, 46E10, 46T05 |
| url | https://arxiv.org/abs/2508.10525 |