Global Dynamics of Monotone Second Order Difference Equation

Abstract

We investigate the global character of the difference equation of the form with several period-two solutions, where f is decreasing in the first variable and is increasing in the second variable. We show that the boundaries of the basins of attractions of different locally asymptotically stable equilibrium solutions or period-two solutions are in fact the global stable manifolds of neighboring saddle or non-hyperbolic equilibrium solutions or period-two solutions. We illustrate our results with the complete study of global dynamics of a certain rational difference equation with quadratic terms.