Patterns of synchrony in lattice dynamical systems

Abstract

From the point of view of coupled systems developed by Stewart, Golubitsky and Pivato, lattice differential equations consist of choosing a phase space Rk for each point in a lattice, and a system of differential equations on each of these spaces Rk such that the whole system is translation invariant. The architecture of a lattice differential equation specifies the sites that are coupled to each other (nearest neighbour coupling (NN) is a standard example). A polydiagonal is a finite-dimensional subspace of phase space obtained by setting coordinates in different phase spaces as equal. There is a colouring of the network associated with each polydiagonal obtained by colouring any two cells that have equal coordinates with the same colour. A pattern of synchrony is a colouring associated with a polydiagonal that is flow-invariant for every lattice differential equation with a given architecture. We prove that every pattern of synchrony for a fixed architecture in planar lattice differential equations is spatially doubly-periodic, assuming that the couplings are sufficiently extensive. For example, nearest and next nearest neighbour couplings are needed for square and hexagonal couplings, but a third level of coupling is needed for the corresponding result to hold in rhombic and primitive cubic lattices. On planar lattices this result is known to fail if the network architecture consists only of NN. The techniques we develop to prove spatial periodicity and finiteness can be applied to other lattices as well.