Global asymptotic stability and trajectory structure rules of high-order nonlinear difference equation

Abstract

In this article, global asymptotic stability and trajectory structure of the following high-order nonlinear difference equation $ z_{n+1} = \frac{z_{n-1}z_{n-2}z_{n-4}+z_{n-1}+z_{n-2}+z_{n-4}+b}{z_{n-1}z_{n-2}+z_{n-1}z_{n-4}+z_{n-2}z_{n-4}+1+b}, \quad n\in N, $ are studied, where $ b\in[0, \infty) $ and the initial conditions $ z_{i}\in(0, \infty), i = 0, -1, -2, -3, -4. $ Using the semi-cycle analysis method, in a prime period, a continuous length of positive and negative semi-cycles of any nontrivial solution appears periodically: 2, 3, 4, 6, 12. Moreover, two examples are given to illustrate the effectiveness of theoretic analysis.

Keywords

• global asymptotic stability
• trajectory structure
• difference equation
• nontrivial solution