Global dynamics of the general second order difference equation in the first quadrant

Abstract

We investigate the global behavior of a general polynomial second order difference equation with non-negative parameters and initial conditions. We establish the relations for local stability of the equilibrium solutions and existence of the period-two solutions. We then use this result to give global behavior results for special ranges of parameters and determine the basins of attraction of all equilibrium and periodic points. We give a class of examples of second order difference equations for which the Julia set can be found explicitly and is represented by a planar curve.