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 +a xnxn−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.  2005 Elsevier Inc. All rights reserved.