Stability of Heteroclinic Cycles: A New Approach Based on a Replicator Equation

Abstract

This paper analyses the stability of cycles within a heteroclinic network formed by six cycles lying in a three-dimensional manifold, for a one-parameter model developed in the context of polymatrix replicator equations. We show the asymptotic stability of the network for a range of parameter values compatible with the existence of an interior equilibrium and we describe an asymptotic technique to decide which cycle (within the network) is visible in numerics. The technique consists of reducing the relevant dynamics to a suitable one-dimensional map, the so-called projective map. The stability of the fixed points of the projective map determines the stability of the associated cycles. The description of this new asymptotic approach is applicable to more general types of networks and is potentially useful in computational dynamics.

Appendices

Appendix A

See Tables 8, 9, 10, 11 and 12.

Table 8 The eigenvalues of system (6) at the vertices, where the entry at line i and row j is the eigenvalue of the vertex $v_j$ in the orthogonal direction to the face $\sigma _i$, and the symbol $*$ means that the vertex $v_i$ does not belong to the face $\sigma _j$ of $[0,1]^3$

Table 9 Eigenvalues of equilibria $B_1$ and $B_2$, depending on $\varvec{\mu }$, at the corresponding faces and pointing to the interior of the cube, where the signs $(-)$, (0), and $(+)$ mean that the eigenvalues are real negative, zero or positive, respectively

Table 10 Branches of $\pi _{\mathcal {S}}$: defining equations of $\Pi _{\xi }$, the matrix of $\pi _\xi$, and their eigenvalues and eigenvectors, for $\xi \in \{ \xi _1, \xi _2, \dots , \xi _6\}$

Table 11 Cycles in $\Pi _{\gamma _5}$: defining equations of $\Pi _{\mathcal {H}}\subset \Pi _{\gamma _5}$, the matrix of $\pi _\mathcal {H}$, and their eigenvalues and eigenvectors, for each $\mathcal {H}\in \{\mathcal {H}_2, \mathcal {H}_4, \mathcal {H}_5, \mathcal {H}_6\}$ in $\Pi _{\gamma _5}$

Table 12 Cycles in $\Pi _{\gamma _8}$: defining equations of $\Pi _{\mathcal {H}}\subset \Pi _{\gamma _8}$, the matrix of $\pi _\mathcal {H}$, and their eigenvalues and eigenvectors, for $\mathcal {H}\in \{\mathcal {H}_1, \mathcal {H}_2, \mathcal {H}_4, \mathcal {H}_5\}$ in $\Pi _{\gamma _8}$

Appendix B. Notation

We list the main notation for constants and auxiliary functions used in this paper in order of appearance with the reference of the section containing their definition Table 13.

Table 13 Notation