Dynamics of Coupled Cell Networks: Synchrony, Heteroclinic Cycles and Inflation

Abstract

We consider the dynamics of small networks of coupled cells. We usually assume asymmetric inputs and no global or local symmetries in the network and consider equivalence of networks in this setting; that is, when two networks with different architectures give rise to the same set of possible dynamics. Focusing on transitive (strongly connected) networks that have only one type of cell (identical cell networks) we address three questions relating the network structure to dynamics. The first question is how the structure of the network may force the existence of invariant subspaces (synchrony subspaces). The second question is how these invariant subspaces can support robust heteroclinic attractors. Finally, we investigate how the dynamics of coupled cell networks with different structures and numbers of cells can be related; in particular we consider the sets of possible "inflations" of a coupled cell network that are obtained by replacing one cell by many of the same type, in such a way that the original network dynamics is still present within a synchrony subspace. We illustrate the results with a number of examples of networks of up to six cells.

Heteroclinic network dynamics on joining coupled cell networks

“…One such example is the occurrence of robust heteroclinic cycles and networks. coupled cell systems that support, in a robust way, simple heteroclinic cycles between two fully synchronised equilibria, see Aguiar et al [2]. For this case, we prove the existence of coupled cell dynamics for the join network supporting a robust simple heteroclinic network with four partially synchronous equilibria.…”

“…Heteroclinic cycles for the N 1 and N 2 admissible equations As shown in [2], there is cell dynamics f 1 for the cells in N 1 such that there are admissible vector fields supporting a robust attracting simple heteroclinic cycle involving p and q. Similarly, there are admissible vector fields for N 2 that support a robust attracting simple heteroclinic cycle involving p and q.…”

“…Thus, a network N with p edge types has p adjacency matrices. Example 2.1 Consider the homogeneous network structure in Figure 1 appearing in [2]. There are two edge types corresponding to the adjacency matrices Note that I 1 = {2, 3}, I 2 = {1, 3} and I 3 = {1, 2}.…”

“…In [2] it is proved the existence of admissible vector fields for the network structure in Figure 1 supporting robust attracting simple heteroclinic cycles involving two fully synchronous equilibria p and q. For example, following the notation of Example 2.4 above, assuming that W u (p) ⊂ {x : x 1 = x 3 }, thus µ 2 = α−β > 0, and assuming that W s (p) ⊂ {x : x 1 = x 2 }, thus µ 3 = α−γ < 0, and the opposite for the invariant manifolds W u (q), W s (q) and corresponding eigenvalues, then there are admissible vector fields f supporting the existence of the heteroclinic cycle depicted in Figure 3.…”

“…Using the same arguments for the local and global construction of heteroclinic orbits presented in Sections 5.2 and 5.3 of [2], that allow to conclude the support by the dynamics of N 1 of a robust attracting simple heteroclinic cycle involving the two equilibria p and q in ∆ 1 with heteroclinic connections in S 12) in the restriction of the join equations (4.9) to the synchrony subspace P 1 × ∆ 2 (∆ 1 × P 2 ) the invariant hyperplanes H p and H q (H p and H q ) are attractors. …”