We describe an approach to the dynamics of coupled systems that is inspired by linear systems theory, transfer function methodology and analog computing. The set-up is broad enough to encompass continuous dynamical systems (ODEs) with general phase spaces, discrete dynamical systems and hybrid systems. We start by describing the idea of dynamical equivalence of network architecture and describe a recent (algorithmic) result that shows that if two network architectures are dynamically equivalent then any system in one architecture can be realized by a system with the second architecture built using the cells of the first system. This turns out to be interesting even if the architectures are the same! We illustrate the combinatorial and constructive aspects of our approach with the example of “inflation” and give some simple examples of how robust heteroclinic cycles and switching networks can occur in small asymmetric networks with low dimensional dynamics as well as how one can use inflation to construct, in a controlled way, networks which admit multiple synchrony subspaces. Parts of the work reported on are the result of collaborations with Nikita Agarwal (Houston), Manuela Aguiar (Porto), Peter Ashwin (Exeter) and Ana Dias (Porto).
Michael Field, professor at University of Houston, Department of Mathematics from 1992, before that at the University of Sydney, Australia, 1976-1992, and the University of Warwick (UK), 1970-1976. Honorary Professor at University of Exeter and Fellow of Institute of Physics. Main areas of research: Symmetric dynamical systems, bifurcation theory, theory of networks, ergodic theory and rates of mixing for flows (supported by NSF continuously since 1994). Books include: “Dynamics and Symmetry” (Imperial College Press, 2007), “Symmetry in Chaos” (with M Golubitsky, 2nd edition, 2009, SIAM).