The Continuations And Bifurcations Of The Equilibria In Two Kinds Of 2-dimensional Coupled Ordinary Differential Equations

In many scientific models, lattices play an important and in some cases essential role, typically modelling an underlying spatial structure in the problem. We mention in particular models arising in chemical reactions [1-2], image processing and pattern recognition [3-7], material science [8], and biology[9-17]. Much theoretical work in lattice differential equations concerns one-dimensional lattices, often with weak coupling between lattice sites. By contrast, we are more concerned with lattice systems in two-dimensions. This is certainly the case in the pattern recognition and material science models above, and as well in the cardiac model[17].In this paper, we compare two kinds of local maps. From the anti-integrability limit, we study the question whether the equilibria at the anti-integrability limit can persist or not with growing coupling coefficient e.Case one: the local map is periodic and has infinitely many zeros. For each equilibrium a at the anti-integrability limit, we define a number b( }. When b( ) < , we find ( ) > 0 and a persists for . But if e is big enough, a undergoes bifurcation.Case two: the local map has finite zeros. In this case, we find the uniform critical value 0 > 0. All the equilibria without coupling persist for . If we impose an additional condition on the local map, we can find < , such there are no new-born equilibria for < 1.We also study in detail the bifurcations of the equilibria for some special local maps.