F. Antoneli, A.P.S. Dias, M. Golubitsky and Y. Wang
Patterns of synchrony in lattice dynamical systems
Nonlinearity.
18 (2005) 2193-2209.
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 is the specification
of which sites are coupled to which (nearest neighbor coupling
is a standard example). A polydiagonal is a finite-dimensional
subspace of phase space obtained by setting coordinates in different
phase spaces equal; a `pattern of synchrony' is 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 neighbor couplings
are needed for square and hexagonal couplings, and 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 nearest
neighbor coupling. The techniques we develop to prove spatial periodicity
and finiteness can be applied to other lattices.