Copyright © 2010 Yunxin Chen and Xianyi Li. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The rule of trajectory structure for fourth-order nonlinear difference equation xn+1=(xan−2+xn−3)/(xan−2xn−3+1), n=0,1,2,…, where a∈[0,1) and the initial values x−3,x−2,x−1,x0∈[0,∞), is described clearly out in this paper. Mainly, the lengths of positive and negative semicycles of its nontrivial solutions are found to occur periodically with prime period 15. The rule is 4+,3−,1+,2−,2+,1−,1+, 1− in a period. By utilizing this rule its positive equilibrium point is verified to be globally asymptotically stable.