Y. Wang and M. Golubitsky
Two-color patterns of synchrony in lattice dynamical systems
Nonlinearity.
18 (2005) 631-657.
Using the theory of coupled cell systems developed by Stewart,
Golubitsky, Pivato, and Torok, we consider patterns of synchrony
in four types of planar lattice dynamical systems: square lattice and hexagonal lattice
differential equations with nearest neighbor coupling and with nearest and next nearest neighbor cou
plings. Patterns of synchrony are flow-invariant subspaces
for all lattice dynamical systems with a given network architecture
that are formed by setting coordinates in different cells equal. Such
patterns can be formed by symmetry (through fixed-point subspaces), but
many patterns cannot be obtained in this way. Indeed, Golubitsky,
Nicol, and Stewart present patterns of synchrony on square
lattices that are not predicted by symmetry. The general theory shows
that finding patterns of synchrony is equivalent to finding balanced
equivalence relations on the set of cells. In a two-color pattern one
set of cells is colored white and the complement black. Two-color patterns in lattice dynamical systems are balanced if the number of white cells connected to a white cell is
the same for all white cells and the number of black cells connected
to a black cell is the same for all black cells. In this paper, we find
all two-color patterns of synchrony of the four kinds of lattice dynamical
systems, and show that all of these patterns, including spatially chaotic
patterns, can be generated from a finite number of distinct patterns. Our
classification shows that all balanced two-colorings in lattices systems
with both nearest and next nearest neighbor couplings are spatially
doubly periodic. We also prove that equilibria associated to each such
two-color pattern can be obtained by codimension one synchrony-breaking
bifurcation from a fully synchronous equilibrium.