Conditions for polynomial Liénard centers

In 1999, Christopher gave a necessary and sufficient condition for polynomial Li′enard centers, which requires coupled functional equations, where the primitive functions of the damping function and the restoring function are involved, to have polynomial solutions. In order to judge whether the coup...

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Bibliographic Details
Published in:Science China. Mathematics 2016-03, Vol.59 (3), p.411-424
Main Authors: Yu, ZhiHeng, Zhang, WeiNian
Format: Article
Language:English
Subjects:
Algebra
Algorithms
Applications of Mathematics
Damping
Mathematical analysis
Mathematical models
Mathematics
Mathematics and Statistics
Origins
Polynomials
Rings (mathematics)
代数关系
充要条件
函数方程
多项式解
恢复功能
既约分解
松
阻尼函数
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Summary:In 1999, Christopher gave a necessary and sufficient condition for polynomial Li′enard centers, which requires coupled functional equations, where the primitive functions of the damping function and the restoring function are involved, to have polynomial solutions. In order to judge whether the coupled functional equations are solvable, in this paper we give an algorithm to compute a Gr¨obner basis for irreducible decomposition of algebraic varieties so as to find algebraic relations among coefficients of the damping function and the restoring function. We demonstrate the algorithm for polynomial Li′enard systems of degree 5, which are divided into 25 cases. We find all conditions of those coefficients for the polynomial Li′enard center in 13 cases and prove that the origin is not a center in the other 12 cases.
ISSN:1674-7283
1869-1862
DOI:10.1007/s11425-015-5100-7