Synchrony in Lattice Differential Equations

Abstract

We survey recent results on patterns of synchrony in lattice differential equations on a square lattice. Lattice differential equations consist of choosing a phase space Rm for each point in a lattice and a system of differential equations on each of these phase spaces such that the whole system is translation invariant. The architecture of a lattice differential equation is the specification of which sites are coupled to which (nearest neighbor coupling is a standard example). A polydiagonal is a finite-dimensional subspace obtained by setting coordinates in different phase spaces equal. A polydiagonal ∆ has k colors if points in ∆ have at most k unequal cell coordinates. A pattern of synchrony is a polydiagonal that is flow-invariant for every lattice differential equation with a given architecture. We survey two main results: the classification of two-color patterns of synchrony and the fact that every pattern of synchrony for a fixed architecture is spatially doubly This work was supported in part by NSF Grant DMS-0244529. The work of FA was supported in part by a FAPESP Grant 03/12631-3. 2 Antoneli, Dias, Golubitsky, Wang periodic assuming that the architecture includes both nearest and next nearest neighbor couplings.