The rule of trajectory structure and global asymptotic stability for a fourth-order rational difference equation

摘要

In this paper, the following fourth-order rational difference equation $$ x_{n+1}=rac{x_n^b +x_{n-2}x_{n -3}^b + a}{x_{n}^bx_{n-2} + x_{n -3}^b + a}, quad n=0, 1, 2, ldots, $$ where $a, b in [0, infty )$ and the initial values $x_{-3},x_{-2}, x_{-1}, x_0 in (0, ; infty )$, is considered and the rule of its trajectory structure is described clearly out. Mainly, the lengths of positive and negative semicycles of its nontrivial solutions are found to occur periodically with prime period 15. The rule is $ 1^+, 1^-, 1^+, 4^-, 3^+, 1^-, 2^+, 2^-$ in a period, by which the positive equilibrium point of the equation is verified to be globally asymptotically stable.