Basin of Attraction through Invariant Curves and Dominant Functions

We study a second-order difference equation of the form z_n+1 = z_n F (z_n-1) + h, where both F (z) and z F (z) are decreasing. We consider a set of invariant curves at h = 1 and use it to characterize the behaviour of solutions when h > 1 and when 0 < h < 1. The case h > 1 is related to the Y2K problem. For 0 < h < 1, we study the stability of the equilibrium solutions and find an invariant region where solutions are attracted to the stable equilibrium. In particular, for certain range of the parameters, a subset of the basin of attraction of the stable equilibrium is achieved by bounding positive solutions using the iteration of dominant functions with attracting equilibria.