Advances in Difference Equations
Volume 2008 (2008), Article ID 678402, 13 pages
doi:10.1155/2008/678402

Dynamical properties for a class of fourth-order nonlinear difference equations

Dongsheng Li1 , Pingping Li1 and Xianyi Li3

1Ministry Education Key Laboratory of Modern Agricultural Equipment and Technology, Jiangsu University, Jiangsu 212013, Zhenjiang, China
 3College of Mathematics and Computational Science, Shenzhen University, Shenzhen 518060, Guangdong, China

Abstract

We consider the dynamical properties for a kind of fourth-order rational difference equations. The key is for us to find that the successive lengths of positive and negative semicycles for nontrivial solutions of this equation periodically occur with same prime period 5. Although the period is same, the order for the successive lengths of positive and negative semicycles is completely different. The rule is …,3+,2−,3+,2−,3+,2−,3+,2−,…, or …,2+,1−,1+,1−,2+,1−,1+,1−,…, or …,1+,4−,1+,4−,1+,4−,1+,4−,…. By the use of the rule, the positive equilibrium point of this equation is proved to be globally asymptotically stable.