Complete heteroclinic networks derived from graphs consisting of two cycles

TL;DR

This work addresses how to realise a given directed graph $G$, specifically the union of two cycles, as a complete heteroclinic network by adding edges (no new vertices) and realizing it with the simplex method. It provides a constructive scheme to obtain complete realisations $G'\supset G$, deriving exact minimal-edge counts in terms of cycle lengths $k$ and $\ell$ and the number of common edges $m$, and shows that each added edge introduces a positive transverse eigenvalue, with the distribution of these eigenvalues depending on the common-edge structure. The authors analyze stability of the resulting cycles via Poincaré return maps and basic transition matrices, giving detailed cases for when there are no common edges or multiple common edges, and discuss how orientation of added edges affects stability. They also explore extensions to graphs with more than two cycles, highlighting limitations and identifying conditions under which complete realisations exist, and discuss potential applications to population dynamics and related systems. Overall, the paper provides explicit, minimal, edge-augmentation techniques to produce complete heteroclinic networks from two-cycle graphs and offers insight into the resulting stability properties of the cycles.

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.

Figures (15)

• Figure 1: A $\Delta$-clique $\Delta_{123}$ in a graph with distribution vertex $\xi_1$.
• Figure 2: A node with two outgoing connections gives rise to a two-dimensional unstable manifold $W^u(\xi_2)$ in the corresponding heteroclinic structure (left). In the middle, $W^u(\xi_2)$ is bounded by being included in the $\Delta$-clique $\Delta_{234}$. On the right, $W^u(\xi_2)$ is bounded by the unstable manifold of an additional equilibrium.
• Figure 3: The graph of the network in Example 1 of PodCasLab2020 is not a tournament (there is no edge between the vertices $\xi_2$ and $\xi_4$) even though the network is complete: only at $\xi_1$ is there a 2-dimensional unstable manifold which is captured by the $\Delta$-clique $\Delta_{123}$.
• Figure 4: Graphs for the Kirk-Silber network (left), the Bowtie (middle), and the House (right).
• Figure 5: For the graph at the top left the simplex method produces and incomplete network since $v_d$ has out-degree 2. An edge can be added connecting $v_{a1}$ to $v_{b1}$ in order to create a $\Delta$-clique to contain the unstable manifold of the distribution node corresponding to $v_d$ (top right). The problem of having out-degree two without a $\Delta$-clique repeats at $v_{a1}$ and can be solved by the addition of an edge $[v_{a2} \to v_{b1}]$ (bottom left). The graph corresponding to the complete network appears at the bottom right.

Theorems & Definitions (18)

• Definition 2.1: Definition 2.1 in GarCas2019
• Proposition 3.1
• proof
• Proposition 3.2
• proof
• Corollary 3.3
• Proposition 4.1
• proof
• Corollary 4.2
• proof