Exotic Patterns of Synchrony in Planar Lattice Networks

Abstract

Patterns of dynamical synchrony that can occur robustly in networks of coupled dynamical systems are associated with balanced colorings of the nodes of the network. In symmetric networks, the orbits of any group of symmetries automatically determine a balanced orbit coloring. Balanced colorings not of this kind are said to be exotic. Exotic colorings occur in infinite planar lattices, both square and hexagonal, with various short-range couplings. In some cases, a balanced two-coloring remains balanced when colors are swapped along suitable diagonals, giving rise to uncountably many distinct exotic colorings. We explain this phenomenon in terms of iterated orbit colorings, in which the quotient of the lattice by an orbit coloring has extra symmetries, allowing new orbit colorings on the quotient, which then lift back to the lattice. We apply the same construction to several other exotic lattice colorings. Two appendices discuss how to modify the notion of balance for networks with diffusive coupling, and how to formalize the differential equations in infinitely many variables that arise for lattices.

Balanced Colorings and Bifurcations in Rivalry and Opinion Networks

“…This was pointed out in [13] for a 12-node bidirectional ring with D 12 symmetry and for certain 2-color lattice patterns. Other examples are discussed in [2,3], and for planar square and hexagonal lattices in [35,38,39]. Here we study networks whose nodes form an m × n array.…”