Global period-doubling bifurcation of quadratic fractional second order difference equation

Abstract

We investigate the local stability and the global asymptotic stability of the difference equation xn+1 = α xn2 + β xn xn-1 + γ xn-1 / A xn2 + B xn xn-1 + C xn-1, n = 0,1,. with nonnegative parameters and initial conditions such that A x n2 + B xn xn-1 + Cxn-1 > 0, for all n ≥ 0. We obtain the local stability of the equilibrium for all values of parameters and give some global asymptotic stability results for some values of the parameters. We also obtain global dynamics in the special case, where β = B = 0, in which case we show that such equation exhibits a global period doubling bifurcation. © 2014 Senada Kalabušić et al.