Two-colour patterns of synchrony in lattice dynamical systems

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