Complete heteroclinic networks derived from graphs consisting of two cycles
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866915748324048896 |
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| author | Castro, Sofia B. S. D. Lohse, Alexander |
| author_facet | Castro, Sofia B. S. D. Lohse, Alexander |
| contents | We address the question how a given connection structure (directed graph) can be realised as a heteroclinic network that is complete in the sense that it contains all unstable manifolds of its equilibria. For a directed graph consisting of two cycles we provide a constructive method to achieve this: (i) enlarge the graph by adding some edges and (ii) apply the simplex method to obtain a network in phase space. Depending on the length of the cycles we derive the minimal number of required new edges. In the resulting network each added edge leads to a positive transverse eigenvalue at the respective equilibrium. We discuss the total number of such positive eigenvalues in an individual cycle and some implications for the stability of this cycle. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_05973 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Complete heteroclinic networks derived from graphs consisting of two cycles Castro, Sofia B. S. D. Lohse, Alexander Dynamical Systems We address the question how a given connection structure (directed graph) can be realised as a heteroclinic network that is complete in the sense that it contains all unstable manifolds of its equilibria. For a directed graph consisting of two cycles we provide a constructive method to achieve this: (i) enlarge the graph by adding some edges and (ii) apply the simplex method to obtain a network in phase space. Depending on the length of the cycles we derive the minimal number of required new edges. In the resulting network each added edge leads to a positive transverse eigenvalue at the respective equilibrium. We discuss the total number of such positive eigenvalues in an individual cycle and some implications for the stability of this cycle. |
| title | Complete heteroclinic networks derived from graphs consisting of two cycles |
| topic | Dynamical Systems |
| url | https://arxiv.org/abs/2501.05973 |