Bifurcations in spatially distributed chains of two-dimensional systems of equations

Abstract

u̇j = Auj + F (uj) + D[uj+1− 2uj + uj−1], j = 1, . . . , N ; u0 ≡ uN , uN+1 ≡ u1. (2) We associate the element uj(t) with the value of a function u(t, xj) of two variables, where xj = 2πjN−1 is the angular coordinate. The main assumption is that the number N of elements in (2) is sufficiently large, so that the parameter ε = 2πN−1 is sufficiently small: 0 < ε ≪ 1. This gives reason to switch from the discrete system (2) to the following system, which is continuous with respect to the spatial variable x: