Dynamics and Bifurcation of a Second Order Quadratic Rational Difference Equation

Abstract

In this paper, we study the dynamics and bifurcation of $$ x_{n+1} = frac{alpha+ beta {x^2}_{n-1}}{A+B {x_n}+C{x^2}_{n-1}}, n=0, 1, 2, ... $$ with positive parameters $alpha, beta, A, B, C, $ and non-negative initial conditions. Among others, we investigate local stability, invariant intervals, boundedness of the solutions, periodic solutions of prime period two and global stability of the positive fixed points.