Combinatorial dynamics

Abstract

Recently Stewart, Golubitsky and co-workers have formulated a general theory of networks of coupled cells. Their approach depends on groupoids, graphs, balanced equivalence relations and ‘quotient networks’. We present a combinatorial approach to coupled cell systems. While largely equivalent to that of Stewart et al., our approach is motivated by ideas coming from analogue computers and avoids abstract algebraic formalism.

Acknowledgements

Thanks to Ian Stewart, Peter Ashwin, Jeroen Lamb and Matthew Nicol for discussions and comments on preliminary versions of this work. Special thanks are due to Philip Holmes, whose comments and critical reading were very helpful in improving the focus and readability of the manuscript, and to the referees whose comments helped improve and extend the ‘translation’ between the combinatorial and groupoid viewpoints. Research supported in part by NSF Grant DMS-0244529.

Notes

We can relax this requirement by allowing for ‘null’ and/or ‘constant’ cells — basically we can either ‘earth’ the unused inputs or feed them a constant input.

But note that our network is not a ‘quotient’ in the sense of Golubitsky et al. An analogy would be that a vector space complement of a subspace

corresponds to the quotient space

.

The term synchrony class corresponds to the ‘polydiagonal subspace’ of Stewart et al. (2003: section 6). Synchrony classes will be ‘robustly polysynchronous’ (Golubitsky et al. 2003: section 4, Stewart et al. 2003: section 6).