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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 861052, 5 pages
http://dx.doi.org/10.1155/2013/861052
Research Article

Limit Cycles and Integrability in a Class of Systems with High-Order Nilpotent Critical Points

School of Science, Linyi University, Shandong, Linyi 276005, China

Received 17 December 2012; Accepted 2 February 2013

Academic Editor: Chuangxia Huang

Copyright © 2013 Feng Li and Jianlong Qiu. 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 polynomial differential systems with high-order nilpotent critical points are investigated in this paper. Those systems could be changed into systems with an element critical point. The center conditions and bifurcation of limit cycles could be obtained by classical methods. Finally, an example was given; with the help of computer algebra system MATHEMATICA, the first 5 Lyapunov constants are deduced. As a result, sufficient and necessary conditions in order to have a center are obtained. The fact that there exist 5 small amplitude limit cycles created from the high-order nilpotent critical point is also proved.