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 $x_{n+1}=\frac{\beta x_n{x}_{n-1}+\gamma x_{n-1}^2+\delta x_n}{B{x}_n{x}_{n-1}+C{x}_{n-1}^2+D{x}_n}$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 $n\ge 0$. More precisely, we consider special cases where either $\gamma =D=0$ or $\beta =D=0$. As we will show both equations exhibit Neimark–Sacker bifurcation, where one of equations ($\gamma =D=0$) probably exhibits Chenciner bifurcation, with two invariant curves, while another Equation ($\beta =D=0$) 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.

Keywords:

• Bifurcation
• difference equation
• global attractivity
• local stability
• Neimark–Sacker

AMS 2020 Mathematics Subject Classifications:

• 37G05
• 37G10
• 39A28
• 39A30

Disclosure statement

No potential conflict of interest was reported by the author(s).