Discrete Dynamics in Nature and Society
Volume 2010 (2010), Article ID 737068, 17 pages
doi:10.1155/2010/737068
Research Article

Uniqueness of Limit Cycles for a Class of Cubic Systems with Two Invariant Straight Lines

Xiangdong Xie,1 Fengde Chen,2 and Qingyi Zhan3

1Department of Mathematics, Ningde Normal University, Ningde, Fujian 352100, China
2School of Mathematics and Computer Science, Fuzhou University, Fuzhou, Fujian 350002, China
3College of Computer and Information Science, Fujian Agriculture and Forestry University, Fuzhou, Fujian 350002, China

Received 1 November 2009; Accepted 12 July 2010

Academic Editor: Binggen Zhang

Copyright © 2010 Xiangdong Xie 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

A class of cubic systems with two invariant straight lines dx/dt=y(1-x2),  dy/dt=-x+δy+nx2+mxy+ly2+bxy2. is studied. It is obtained that the focal quantities of O(0,0) are, W0=δ; if W0=0, then W1=m(n+l); if W0=W1=0, then W2=−nm(b+1); if W0=W1=W2=0, then O is a center, and it has been proved that the above mentioned cubic system has at most one limit cycle surrounding weak focal O(0,0). This paper also aims to solve the remaining issues in the work of Zheng and Xie (2009).