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

In this article, global asymptotic stability and trajectory structure of the following high-order nonlinear difference equation

$$z_{n+1}=\frac{z_{n-1}{z}_{n-2}{z}_{n-4}+z_{n-1}+z_{n-2}+z_{n-4}+b}{z_{n-1}{z}_{n-2}+z_{n-1}{z}_{n-4}+z_{n-2}{z}_{n-4}+1+b},n\in N,$$

are studied, where $b\in [0,\mathrm{\infty })$ and the initial conditions $z_i\in (0,\mathrm{\infty }),i=0,-1,-2,-3,-4.$ 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.