Qualitative properties for a fourth-order rational difference equation

In this paper, we use a method different from the known literature to investigate the qualitative properties of the following fourth-order rational difference equation: xn+1=xnxn−1xn−3+xn+xn−1+xn−3+axnxn−1+xnxn−3+xn−1xn−3+1+a,n=0,1,2,…, where a∈[0,∞) and the initial values x−3,x−2,x−1,x0∈(0,∞). The successive lengths of positive and negative semicycles of nontrivial solutions of the above equation is found to periodically occur, that is, …,3+,2−,1+,1−,3+,2−,1+,1−,3+,2−,1+,1−,3+,2−,1+,1−,…, or, …,2+,1−,1+,3−,2+,1−,1+,3−,2+,1−,1+,3−,2+,1−,1+,3−,2+,1−,1+,3−,…. By using the rule, the positive equilibrium point of the equation is verified to be globally asymptotically stable.