Analysis of Trajectory Structure and GAS for a High-Order Nonlinear Difference Equation

This article delves into the trajectory structure rules of a specific fifth-order rational difference equation:$$s_{m+1}=\frac{s_ms_{m-2}s_{m-3}s_{m-4}+s_ms_{m-2}+s_ms_{m-3}+s_{m-2}s_{m-3}+s_{m-4}+a}{s_ms_{m-2}s_{m-3}+s_ms_{m-2}s_{m-4}+s_ms_{m-3}s_{m-4}+s_{m-2}s_{m-3}s_{m-4}+1+a}$$where the initial conditions satisfy $s_i\in (0,\infty)$, $i=-4,-3,-2,-1,0$, and the parameters $a\in [0,\infty).$ As the initial values vary, the lengths of consecutive positive and negative semi-cycles for non-trivial solutions exhibit a periodic pattern with a prime period of 31. The rule within one period is $1^-, 2^+, 1^-, 1^+, 1^-, 1^+, 2^-, 4^+, 3^-, 2^+, 2^-, 1^+, 5^-, 1^+, 1^-,$ $ 3^+ $. Through the application of this rule,the global asymptotic stability(GAS) of the positive fixed point of the equation is proven. In the end, three instances are utilized to demonstrate the accuracy of the theoretical conclusions.