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Coupled cell networks - interplay between architecture & dynamics

We are interested in studying the dynamics on coupled cell networks using the network topology or architecture. The original motivation of this work comes from the work by Aguiar et al. [4] on dynamics of coupled cell networks and Dias and Stewart [18] on equivalence of coupled cell networks with linear phase space. In this thesis, we give a necessary and sufficient condition for the dynamical equivalence of two coupled cell networks. The results are applicable to both continuous and discrete dynamical systems and are framed in terms of what we call input and output equivalence. We also give an algorithm that allows explicit construction of the cells in a system with a given network architecture in terms of the cells from an equivalent system with different network architecture. We provide a number of examples to illustrate the results we obtain. The dynamics of large coupled dynamical systems can be extremely difficult to analyze. One way of approaching the study of dynamics of networks is to start with a simple, well-understood but interesting small network and investigate how the simple network can be naturally embedded in a larger network in such a way that the dynamics of the small network appears in the dynamics of the larger network. This process is termed inflation, and was introduced by Aguiar et al. [4]. We give a necessary and sufficient condition for the existence of a strongly connected inflation. We also provide a simple algorithm to construct the inflation as a sequence of simple inflations. Finally, we give an example of a 3-cell coupled cell network having non-trivial synchrony subspaces that supports heteroclinic cycles with switching between them.