Dynamics and behavior of the higher order rational difference equation (Q2792546)

The author investigates periodicity and global stability of solutions of the difference equation NEWLINE\[NEWLINE x_{n+1}=ax_n+bx_{n-k}+cx_{n-l}-\frac{dx_{n-s}}{ex_{x-s}-\alpha x_{n-t}},\tag{\(*\)} NEWLINE\]NEWLINE where \(a,b,c,d,e,\alpha\) are positive real numbers. Typical results given in the paper are the following statements.NEWLINENEWLINETheorem. (i) The equilibrium point \(\bar x=\frac{d}{(\alpha-e)(1-a-b-c)}\) of Equation \((*)\) is locally asymptotically stable if \(\alpha-e>2d\). (ii) The equilibrium point \(\bar x\) is a global attractor of Equation \((*)\) if \(a+b+c<1\).NEWLINENEWLINETheoretical results are illustrated by a number of numerical examples.