Narrow Results

A coupled cell network is a schematic diagram employed to define a class of differential equations, and can be thought of as a directed graph whose nodes (cells) represent dynamical systems and whose edges (arrows) represent couplings. Often the nodes and edges are labeled to distinguish different types of system and coupling. The associated differential equations reflect this structure in a natural manner. The network is homogeneous if there is one type of cell and one type of arrow, and moreover, every cell lies at the head end of the same number r of arrows. This number is the valency of the network. We use a group-theoretic formula usually but incorrectly attributed to William Burnside to enumerate homogeneous coupled cell networks with N cells and valency r, in both the disconnected and connected cases. We compute these numbers explicitly when N, r ≤ 6.

Equivariant dynamical systems possess canonical flow-invariant subspaces, the fixed-point spaces of subgroups of the symmetry group. These subspaces classify possible types of symmetry-breaking. Coupled cell networks, determined by a symmetry groupoid, also possess canonical flow-invariant subspaces, the balanced polydiagonals. These subspaces classify possible types of synchrony-breaking, and correspond to balanced colorings of the cells. A class of dynamical systems that is common to both theories comprises networks that are symmetric under the action of a group Γ of permutations of the nodes ("cells"). We investigate connections between balanced polydiagonals and fixed-point spaces for such networks, showing that in general they can be different. In particular, we consider rings of ten and twelve cells with both nearest and next-nearest neighbor coupling, showing that exotic balanced polydiagonals — ones that are not fixed-point spaces — can occur for such networks. We also prove the "folk theorem" that in any Γ-equivariant dynamical system on R^k the only flow-invariant subspaces are the fixed-point spaces of subgroups of Γ.

This paper continues the study of patterns of synchrony (equivalently, balanced colorings or flow-invariant subspaces) in symmetric coupled cell networks, and their relation to fixed-point spaces of subgroups of the symmetry group. Our aim is to provide a group-theoretic explanation of the "exotic" balanced coloring previously discussed in Part 2. Here we show that the pattern can be obtained as a projection into two dimensions of a fixed-point pattern in a three-dimensional lattice. We prove a general theorem giving sufficient conditions for such a construction to lead to a balanced coloring, for an arbitrary direct product of group networks.

Following Golubitsky, Stewart, and others, we give definitions of networks and input trees. In order to make our work as general as possible, we work with a somewhat extended notion of multiplicity, and introduce the concept of "bunching" of trees. We then define balanced equivalence relations on networks, and a partial ordering on these relations. Previous work has shown that there is a maximal balanced equivalence relation on networks of certain classes: we provide a different style of proof which gives this result for any network. We define two algorithms to determine this relation in practice on a given finite network — one for use with networks with all multiplicities equal, and a second for the more general case. We then provide illustrative examples of each algorithm in use. We show both of these algorithms to be quartic in the size of the given network.