Global behavior for a fourth-order rational difference equation

In this paper, we use a method different from the known literature to investigate the global behavior of the following fourth-order rational difference equation: xn+1=xn−1xn−2xn−3+xn−1+xn−2+xn−3+axn−1xn−2+xn−1xn−3+xn−2xn−3+1+a,n=0,1,2,…, where a∈[0,∞) and the initial values x−3, x−2, x−1, x0∈(0,∞). The rule of the trajectory structure for the solutions of the equation is clearly described out. The successive lengths of positive and negative semicycles of nontrivial solutions of the above equation is found to periodically occur. However, the order for them to occur is completely different although there is the same prime period 7. The rule is 3+, 1−, 1+, 2− or 3−, 1+, 1−, 2+ in a period. By using the rule, the positive equilibrium point of the equation is verified to be globally asymptotically stable.