Enumerating periodic patterns of synchrony via finite bidirectional networks

Abstract

Periodic patterns of synchrony are lattice networks whose cells are coloured according to a local rule, or balanced colouring, and such that the overall system has spatial periodicity. These patterns depict the finite-dimensional flow-invariant subspaces for all the lattice dynamical systems, in the given lattice network, that exhibit those periods. Previous results relate the existence of periodic patterns of synchrony, in n-dimensional Euclidean lattice networks with nearest neighbour coupling architecture, with that of finite coupled cell networks that follow the same colouring rule and have all the couplings bidirectional. This paper addresses the relation between periodic patterns of synchrony and finite bidirectional coloured networks. Given an n-dimensional Euclidean lattice network with nearest neighbour coupling architecture, and a colouring rule with k colours, we enumerate all the periodic patterns of synchrony generated by a given finite network, or graph. This enumeration is constructive and based on the automorphisms group of the graph.

Meadows and the equational specification of division

“…For an arbitrary model of E ftc−cm we prefer to use generalised common meadow, or GCM, in an attempt to be concise and avoid overly logical jargon. We mention that in [21] and [22] the original terminology of [8] is used.…”

“…GCMs allow a rich world of homomophisms which has first been explored in general by Dias and Dinis in [21], and in the context of finite GCMs in [22], where however, homomorphisms are not required to respect division. Below we will insist that a GCM-homomorphism respects division as well.…”

“…Proof. A model of E ftc−cm + AVL in which φ 2,3,5 fails can be found following the structure theory for GCM's as set out in [21,22]. The underlying idea is that all GCM's M , contain a lattice L M made up by elements from 0 •M .…”