Two-colour patterns of synchrony in lattice dynamical systems

Abstract

Using the theory of coupled cell systems developed by Stewart, Golubitsky, Pivato and Török, we consider patterns of synchrony in four types of planar lattice dynamical systems: square lattice and hexagonal lattice differential equations with nearest neighbour coupling and with nearest and next nearest neighbour couplings. 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 lattice 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-colour pattern one set of cells is coloured white and the complement black. Two-colour 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-colour patterns of synchrony of the four kinds of lattice dynamical systems, and show that all of these patterns, including spatially complicated patterns, can be generated from a finite number of distinct patterns. Our classification shows that all balanced two-colourings in lattice systems with both nearest and next nearest neighbour couplings are spatially doubly periodic. We also prove that equilibria associated with each such two-colour pattern can be obtained by codimension one synchrony-breaking bifurcation from a fully synchronous equilibrium.