F. Antoneli, A.P.S. Dias, M. Golubitsky and Y. Wang
Flow invariant subspaces for lattice dynamical systems
In: Bifurcation Theory and Spatio-Temporal Pattern Formation.
(W. Nagata and N.S. Namachchivaya, eds.)
Fields Institute Communications, 2006, 1-8.
Stewart et al. have shown that flow invariant subspaces for
coupled networks are equivalent to a combinatorial notion of
a balanced coloring. Wang and Golubitsky have classified
all balanced two colorings of planar lattices with either
nearest neighbor (NN) or both nearest neighbor and next nearest
neighbor coupling (NNN). This classification gives a rich
set of patterns and shows the existence of many nonspatially
periodic patterns in the NN case. However, all balanced
two-colorings in the NNN case on the square and hexagonal
lattices are spatially periodic. We survey these and new
results showing that all balanced k-colorings in the NNN
case on square lattices are spatially periodic.