Robust heteroclinic cycles in coupled cell systems: identical cells with asymmetric inputs

Abstract

We consider the dynamics of coupled cells connected so that the network may or may not have symmetries. 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 different structures of the network relate to and force the appearance of invariant subspaces (synchrony subspaces) associated with partitions (synchrony classes) for the network. The second question is how these invariant subspaces can allow the appearance of robust heteroclinic attractors in such networks even if their symmetries are not sufficient to force their appearance. The third question is how the dynamics of coupled cell networks of 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, such 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.

preprint (February 2008).

Professor Mike Field
Department of Mathematics
University of Houston
Houston, TX 77204-3008