On the Recursive Sequence xn+1=axn−1b+cxnxn−1

In this paper, we investigate the dynamical behaviors of the rational difference equation xn=(axn−1)/(b+cxnxn−1) with arbitrary initial conditions, where a, b, and c are real numbers. A general solution is obtained. The asymptotic stability of the equilibrium points is investigated, using a nonlinear stability criterion combined with basin of attraction analysis and simulation to determine the stability regions of the equilibrium points. The existence of the periodic solutions is discussed. We investigate the codim-1 bifurcations of the equation. We show that the equation exhibits a Neimark–Sacker bifurcation. For this bifurcation, the topological normal form is computed. To confirm our theoretical results, we performed a numerical simulation as well as numerical bifurcation analysis by using the Matlab package MatContM.