Spatially Periodic Patterns of Synchrony in Lattice Networks

We consider n-dimensional Euclidean lattice networks with nearest neighbour 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 colours, and are described by a local colouring rule, named balanced colouring. Previous results with planar lattices show that patterns of synchrony can exhibit several behaviours like periodicity. Considering sufficiently extensive couplings, spatial periodicity appears for all the balanced colourings with k colours. However, there is not a direct way of relating the local colouring rule and the colouring of the whole lattice network. We state a necessary and sufficient condition for the existence of a spatially periodic pattern of synchrony, given an n-dimensional lattice network with nearest neighbour couplings, and a local colouring rule with k colours. As an intermediate step, we obtain the proportion of the cells for each colour, for the lattice network and any finite bidirectional network with the same balanced colouring.