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Journal of Function Spaces
Volume 2015, Article ID 859015, 7 pages
Research Article

Limit Cycles and Analytic Centers for a Family of Degree Systems with Generalized Nilpotent Singularities

1School of Mathematics and Statistics, Henan University of Science and Technology, Luoyang, Henan 471023, China
2College of Mathematical Science, Luoyang Normal University, Luoyang, Henan 471022, China
3Guizhou Key Laboratory of Economics System Simulation, Guizhou University of Finance and Economics, Guiyang, Guizhou 550004, China

Received 25 July 2014; Accepted 11 September 2014

Academic Editor: Józef Banaś

Copyright © 2015 Yusen Wu 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.


With the aid of computer algebra system Mathematica 8.0 and by the integral factor method, for a family of generalized nilpotent systems, we first compute the first several quasi-Lyapunov constants, by vanishing them and rigorous proof, and then we get sufficient and necessary conditions under which the systems admit analytic centers at the origin. In addition, we present that seven amplitude limit cycles can be created from the origin. As an example, we give a concrete system with seven limit cycles via parameter perturbations to illustrate our conclusion. An interesting phenomenon is that the exponent parameter controls the singular point type of the studied system. The main results generalize and improve the previously known results in Pan.