F. Antoneli, A.P.S. Dias, M. Golubitsky and Y. Wang
Synchrony in lattice differential equations
In: Some Topics In Industrial and Applied Mathematics.
(R. Jeltsch, T. Li, and I. Sloan, eds.)
Contemporary Applied Mathematics Series 8
World Scientific Publ. Co., 2007,
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 R^m 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 Delta has
k colors if points in Delta 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 periodic assuming that the architecture includes both nearest and next
nearest neighbor couplings.