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 flowinvariant 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.

Enumerating periodic patterns of synchrony via finite bidirectional networks

“…Lemma 3.1 presents a summary of results in Dias & Pinho (2009) (i) (Dias & Pinho 2009, lemma 4.2) The adjacency matrix B of the network has the following block structure: …”

“…By Dias & Pinho (2009, theorem 5.3), this ensures the existence of a function Π : L −→ C with the property Π (l + g) = σ g (Π(l)), where σ g = Φ(g), for all g, l ∈ L. By the proof of theorem 5.4 in Dias & Pinho (2009), it follows that there is a periodic balanced colouring ξ L with the colouring matrix A and such that condition (4.1) is verified.…”

“…First suppose that there is a periodic balanced colouring ξ L , with the colouring matrix A, and denote byL the set of periods of ξ L . Then, by the proof of theorem 5.4 in Dias & Pinho (2009), there is a finite IEHB network as described and, in particular, there is one IEHB network with #C = |L/L|, the number of cells in a fundamental domain of ξ L .…”

“…The restrictions imposed by the fixed colouring rule on the finite bidirectional networks, based on Dias & Pinho (2009), are summarized in §3. We define root networks in §3a, after the statement of important results on the decompositions of a finite bidirectional network, or graph, and on the relation between the possible decompositions and the automorphisms group of the graph (see proposition 3.9).…”

“…In §4, we describe the procedures to generate periodic patterns of synchrony from finite bidirectional networks, derived by Dias & Pinho (2009). These procedures depend on suitable decompositions of the graph that corresponds to the finite bidirectional network.…”