Basin of attraction through invariant curves and dominant functions (Q1723208)

Summary: 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.