In this paper we study the asymptotic behaviour of the positive solutions of the system of two difference equations where are positive constants and the initial values are positive numbers.

Advances in Difference Equations is a peer-reviewed open access journal published under the brand SpringerOpen. The theory of difference equations, the methods used, and their wide applications have advanced beyond their adolescent stage to occupy a central position in applicable analysis. In fact, in the last 12 years, the proliferation of the subject has been witnessed by hundreds of research articles, several monographs, many international conferences, and numerous special sessions. The theory of differential and difference equations forms two extreme representations of real world problems. For example, a simple population model when represented as a differential equation shows the good behavior of solutions whereas the corresponding discrete analogue shows the chaotic behavior. The actual behavior of the population is somewhere in between. Articles published in Advances in Difference Equations will include such situations. The aim of Advances in Difference Equations is to report new developments in the field of difference equations, and their applications in all fields. Advances in Difference Equations will accept high-quality articles containing original research results and survey articles of exceptional merit.

In this paper, we study the stability of the zero equilibria of two close-to-symmetric systems of difference equations with exponential terms in the special case in which one of their eigenvalues is equal to $-1$ and the other eigenvalue has an absolute value of less than 1. In the present study, we use the approach of center manifold theory.