Global behaviour of solutions to a class of second-order rational difference equations when prime period-two solutions exist

Abstract

For non-negative parameters α, β, γ, A, B and C such that A+B+C>0, consider the class of difference equations where either if A = 0, or R = [0,∞)2 if A>0. This class consists of 49 non-trivial second-order rational difference equations. We give a complete qualitative description of the global behaviour of solutions including basins of attraction of equilibria and prime period-two (PP2) solutions for all nonlinear difference equations (E) for which PP2 solutions exist. In particular, we show that every solution of (E) converges to an equilibrium or to a PP2 solution.