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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 398609, 5 pages
http://dx.doi.org/10.1155/2013/398609
Research Article

Fourteen Limit Cycles in a Seven-Degree Nilpotent System

1Guangxi Key Laboratory of Trusted Software, School of Computing Science and Mathematics, Guilin University of Electronic Technology, Guilin 541004, China
2Department of Mathematics, Hezhou University, Hezhou 542800, China

Received 13 August 2013; Accepted 30 October 2013

Academic Editor: Isaac Garcia

Copyright © 2013 Wentao Huang 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

Center conditions and the bifurcation of limit cycles for a seven-degree polynomial differential system in which the origin is a nilpotent critical point are studied. Using the computer algebra system Mathematica, the first 14 quasi-Lyapunov constants of the origin are obtained, and then the conditions for the origin to be a center and the 14th-order fine focus are derived, respectively. Finally, we prove that the system has 14 limit cycles bifurcated from the origin under a small perturbation. As far as we know, this is the first example of a seven-degree system with 14 limit cycles bifurcated from a nilpotent critical point.