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

We investigate the Neimark–Sacker bifurcation of the equilibrium of two special cases of the difference Equation xn+1=βxnxn−1+γxn−12+δxnBxnxn−1+Cxn−12+Dxnwhere 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 x0 are arbitrary nonnegative numbers such that Bxnxn−1+Cxn−12+Dxn>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.