Spatially Periodic Patterns of Synchrony in Lattice Networks

Abstract

We consider n-dimensional Euclidean lattice networks with nearest neighbor coupling architecture. The associated lattice dynamical systems are infinite systems of ordinary differential equations, the cells, indexed by the points in the lattice. A pattern of synchrony is a finite-dimensional flow-invariant subspace for all lattice dynamical systems with the given network architecture. These subspaces correspond to a classification of the cells into k classes, or colors, and are described by a local coloring rule, named balanced coloring. Previous results with planar lattices show that patterns of synchrony can exhibit several behaviors such as periodicity. Considering sufficiently extensive couplings, spatial periodicity appears for all the balanced colorings with k colors. However, there is not a direct way of relating the local coloring rule and the coloring of the whole lattice network. Given an n-dimensional lattice network with nearest neighbor coupling architecture, and a local coloring rule with k colors, we state a necessary and sufficient condition for the existence of a spatially periodic pattern of synchrony. This condition involves finite coupled cell networks, whose couplings are bidirectional and whose cells are colored according to the given rule. As an intermediate step, we obtain the proportion of the cells for each color, for the lattice network and any finite bidirectional network with the same balanced coloring. A crucial tool in obtaining our results is a classical theorem of graph theory concerning the factorization of even degree regular graphs, a class of graphs where lattice networks are included.