Neimark–Sacker bifurcation of two second-order rational difference equations

Abstract

We investigate the Neimark–Sacker bifurcation of the equilibrium of two special cases of the difference Equation x n + 1 = β x n x n − 1 + γ x n − 1 2 + δ x n B x n x n − 1 + C x n − 1 2 + D x n where the parameters β, γ, δ, B, C, D are non-negative numbers which satisfy B + C + D>0 and the initial conditions x − 1 and x 0 are arbitrary nonnegative numbers such that B x n x n − 1 + C x n − 1 2 + D x n > 0 for all n ≥ 0 . More precisely, we consider special cases where either γ = D = 0 or β = D = 0 . As we will show both equations exhibit Neimark–Sacker bifurcation, where one of equations ( γ = D = 0 ) probably exhibits Chenciner bifurcation, with two invariant curves, while another Equation ( β = D = 0 ) exhibits simple Neimark–Sacker bifurcation with one invariant curve. We will also obtain some global asymptotic stability result for each equation in the subset of the parametric region of local stability.