Global dynamics of perturbation of certain rational difference equation

Authors: SABINA HRUSTIC, MUSTAFA KULENOVIC, SAMRA MORANJKIC, ZEHRA NURKANOVIC

Abstract: We investigate the global asymptotic stability of the difference equation of the form \begin{equation*} x_{n+1}=\frac{A x_{n}^{2}+F}{a x_{n}^{2}+e x_{n-1}}, \quad n=0,1,\ldots, \end{equation*}% with positive parameters and nonnegative initial conditions such that $x_0 + x_{-1}>0$. The map associated to this equation is always decreasing in the second variable and can be either increasing or decreasing in the first variable depending on the parametric space. In some cases, we prove that local asymptotic stability of the unique equilibrium point implies global asymptotic stability.

Keywords: Difference equation, attractivity, invariant, period doubling bifurcation, periodic solutions

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