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 \[ x_{n+1}=\frac{\beta x_n x_{n-1}+ \gamma x_{n-1}^2 +\delta x_n}{B x_n x_{n-1}+C x_{n-1}^2 +D x_n} \]xn+1=βxnxn−1+γxn−12+δxnBxnxn−1+Cxn−12+Dxn 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} $ x−1 and $ x_0 $ x0 are arbitrary nonnegative numbers such that $ B x_n x_{n-1}+C x_{n-1}^2 +D x_n \gt 0 $ Bxnxn−1+Cxn−12+Dxn>0 for all $ n \geq 0 $ n≥0. More precisely, we consider special cases where either $ \gamma =D=0 $ γ=D=0 or $ \beta =D=0 $ β=D=0. As we will show both equations exhibit Neimark–Sacker bifurcation, where one of equations ( $ \gamma =D=0 $ γ=D=0) probably exhibits Chenciner bifurcation, with two invariant curves, while another Equation ( $ \beta =D=0 $ 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.