M. Golubitsky, M. Nicol and I. Stewart
Some curious phenomena in coupled cell systems
J. Nonlinear Sci.
14 (2) (2004) 207-236.
We consider several examples of coupled cell networks with synchronous dynamics
that are unexpected from symmetry considerations but are natural using a theory
developed by Stewart, Golubitsky, and Pivato. In particular we demonstrate
patterns of synchrony in networks with small numbers of cells and in lattices
(and periodic arrays) of cells that cannot readily be explained by
conventional symmetry considerations. We also show that different types of
dynamics can coexist robustly in single
solutions of systems of coupled identical cells. The examples we consider
include a 3-cell system exhibiting equilibria, periodic, and quasiperiodic
states in different cells; periodic 2n X 2n arrays of cells that generate
2^n different random patterns of synchrony from one symmetry generated
solution; and systems exhibiting multirhythms (periodic solutions with
rationally related periods in different cells). Our theoretical results
include the observation that reduced equations on a center manifold of a
skew-product system inherit a skew-product form.