## 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.

preprint (January 2009).

For (p)reprint,
e-mail: mikefield@gmail.com or Michael.J.Field@rice.edu

Professor Mike Field

Department of Mathematics

Rice University

6100 S Main St

Houston

TX 77005-1892