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arXiv · 2501.05973

Complete heteroclinic networks derived from graphs consisting of two cycles

Sofia B. S. D. Castro·Alexander Lohse
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Abstract

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.

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Sofia B. S. D. Castro, Alexander Lohse. 2025-01-10. Complete heteroclinic networks derived from graphs consisting of two cycles. https://arxiv.org/abs/2501.05973

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