Flow Invariant Subspaces for Lattice Dynamical Systems

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. 2000 Mathematics Subject Classification. Primary: 37C10, 37C20; Secondary: 34A34. This research was supported in part by NSF Grant DMS-0244529 and ARP Grant 0036520032-001. The work of FA was supported in part by a grant from FAPESP. c ©0000 American Mathematical Society