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

Dynamical Properties for a Class of Fourth-Order Nonlinear Difference Equations

Dongsheng Li,1,2 Pingping Li,1 and Xianyi Li3

1Ministry Education Key Laboratory of Modern Agricultural Equipment and Technology, Jiangsu University, Jiangsu 212013, Zhenjiang, China
2School of Economics and Management, University of South China, Hunan 421001, Hengyang, China
3College of Mathematics and Computational Science, Shenzhen University, Shenzhen 518060, Guangdong, China

Received 6 May 2007; Revised 14 August 2007; Accepted 18 September 2007

Academic Editor: Jianshe Yu

Copyright © 2008 Dongsheng Li et al. 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

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.