Boundedness of solutions and stability of certain second-order difference equation with quadratic term

Abstract

Abstract We consider the second-order rational difference equation xn+1=γ+δxnxn−12, $$ {x_{n+1}=\gamma +\delta \frac{x_{n}}{x^{2}_{n-1}}}, $$ where γ, δ are positive real numbers and the initial conditions x−1 $x_{-1}$ and x0 $x_{0}$ are positive real numbers. Boundedness along with global attractivity and Neimark–Sacker bifurcation results are established. Furthermore, we give an asymptotic approximation of the invariant curve near the equilibrium point.

Keywords

• Difference equation
• Birkhoff normal form
• Boundedness
• Invariant set
• Neimark–Sacker bifurcation
• Stability