INSTITUTE OF PHYSICS PUBLISHING NONLINEARITY Nonlinearity 18 (2005) 2193–2209 doi:10.1088/0951-7715/18/5/016 Patterns of synchrony in lattice dynamical systems Fernando Antoneli 1 , Ana Paula S Dias 2 , Martin Golubitsky 3 and Yunjiao Wang 3 1 Department of Applied Mathematics, University of S˜ ao Paulo, S˜ ao Paulo, SP 05508-090, Brazil 2 Centro de Matem´ atica, Universidade do Porto, Porto, 4169-007, Portugal 3 Department of Mathematics, University of Houston, Houston, TX 77204-3008, USA Received 16 December 2004, in final form 6 May 2005 Published 1 July 2005 Online at stacks.iop.org/Non/18/2193 Recommended by J Lega Abstract From the point of view of coupled systems developed by Stewart, Golubitsky and Pivato, lattice differential equations consist of choosing a phase space R k for each point in a lattice, and a system of differential equations on each of these spaces R k 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. Mathematics Subject Classification: 34C99, 37G99, 82B20 1. Introduction Many physical and biological systems can be modelled by networks of systems of differential equations. Networks of differential equations possess additional structure, namely, canonical observables—the dynamical behaviour of the individual network nodes [18]. These 0951-7715/05/052193+17$30.00 © 2005 IOP Publishing Ltd and London Mathematical Society Printed in the UK 2193