Synchrony in Lattice Differential Equations [chapter]

Fernando Antoneli, Ana Paula S. Dias, Martin Golubitsky, Yunjiao Wang
2007 Series in Contemporary Applied Mathematics  
We survey recent results on patterns of synchrony in lattice differential equations on a square lattice. Lattice differential equations consist of choosing a phase space R m for each point in a lattice and a system of differential equations on each of these phase 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
more » ... e). A polydiagonal is a finite-dimensional subspace obtained by setting coordinates in different phase spaces equal. A polydiagonal ∆ has k colors if points in ∆ have at most k unequal cell coordinates. A pattern of synchrony is a polydiagonal that is flow-invariant for every lattice differential equation with a given architecture. We survey two main results: the classification of two-color patterns of synchrony and the fact that every pattern of synchrony for a fixed architecture is spatially doubly * periodic assuming that the architecture includes both nearest and next nearest neighbor couplings.
doi:10.1142/9789812709356_0003 fatcat:odh7455hofcrvbj2em4txrklom