Global Dynamics and Bifurcations of Certain Second Order Rational Difference Equation with Quadratic Terms

We investigate global dynamics of the equation x n + 1 = x n - 1 a x n 2 + e x n - 1 + f , n = 0 , 1 , 2 , … , where the parameters a , e and f are nonnegative numbers with condition a + e + f > 0 and the initial conditions x - 1 , x 0 are arbitrary nonnegative numbers such that x - 1 + x 0 > 0 . The global dynamics of this equation consists of three bifurcations, two exchange of stability bifurcations and one global period doubling bifurcation.