An Isaac Newton Institute Workshop

Theory and Applications of Coupled Cell Networks

Patterns of Synchrony in Lattice Dynamical Systems

28th September 2005

Authors: Fernando Antoneli (Sao Paulo), Ana Paula S. Dias (Porto), Martin Golubitsky (Houston), Yunjiao Wang (Houston)

Abstract

>From the point of view of coupled systems developed by Stewart, Golubitsky, and Pivato, lattice differential equations consist of choosing a phase space for each point in a lattice and a system of differential equations on each of these spaces such that the whole system is translation invariant. The architecture of a lattice differential equation is the specification of which sites are coupled to which (nearest neighbor coupling is a standard example). A polydiagonal is a finite-dimensional subspace of phase space obtained by setting coordinates in different phase spaces equal. There is a coloring of the network associated to each polydiagonal that is obtained by coloring any two cells that have equal coordinates with the same color. A pattern of synchrony is a coloring associated to a polydiagonal that is flow-invariant for every lattice differential equation with a given architecture. We prove that every pattern of synchrony for a fixed architecture in planar lattice differential equations is spatially doubly periodic assuming that the couplings are sufficiently extensive. For example, nearest and next nearest neighbor couplings are needed for square and hexagonal couplings, and a third level of coupling is needed for the corresponding result to hold in rhombic and primitive cubic lattices. On planar lattices this result is known to fail if the network architecture consists only of nearest neighbor coupling. The techniques we develop to prove spatial periodicity and finiteness can be applied to other lattices.

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