Basin of Attraction through Invariant Curves and Dominant Functions

Abstract

We study a second-order difference equation of the formzn+1=znF(zn-1)+h, where bothF(z)andzF(z)are decreasing. We consider a set of invariant curves ath=1and use it to characterize the behaviour of solutions whenh>1and when0<h<1. The caseh>1is related to the Y2K problem. For0<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.