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Complete Study on a Bi-Center Problem for the Z2-Equivariant Cubic Vector Fields

Acta Mathematica Sinica,English Series 2011年第7期27卷 1379-1394页

作者： Yi Rong LIU Ji Bin LI Department ofMathematics Zhejiang Normal University Jinhua 321004 P. R. China Department of Mathematics Zhejiang Normal University Jinhua 321004 P. R. China School of Science Kunming University of Science and Technology Kunming 650093 P. R. China

For the planar Z2-equivariant cubic systems having twoelementary focuses, the characterization of a bi-center problem and shortened expressions of the first six Liapunov constants are completely discussed. The necessary and sufficient conditions for the existence of the bi-center are obtained. All possible first integrals are given. Under small Z2-equivariant cubic perturbations, the conclusion that there exist at most 12 small-amplitude limit cycles with the scheme （6 II 6） is proved.

关键词： Center problem Liapunov constant focal value integral factor invariant integral cubic polynomial system

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Center, Limit Cycles and Isochronous Center of a Z_4-equivariant Quintic System

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Acta Mathematica Sinica,English Series 2010年第6期26卷 1183-1196页

作者： Chao Xiong DU Hei Long MI Yi Rong LIU Department of Hu'nan Shaoyang University Shaoyang 422000 P. R. China and School of Mathematics Central South University Changsha 410083 P. R. China Department of Information Science Hu'nan Commercial University Changsha 410083 P. R. China School of Mathematics Central South University Changsha 410083 P. R. China

In this paper, we study the limit cycles bifurcations of four fine focuses in Z4-equivariant vector fields and the problems that its four singular points can be centers and isochronous centers at the same time. By computing the Liapunov constants and periodic constants carefully, we show that for a certain Z4-equivariant quintic systems, there are four fine focuses of five order and five limit cycles can bifurcate from each, we also find conditions of center and isochronous center for this system. The process of proof is algebraic and symbolic by using common computer algebra soft such as Mathematica, the expressions after being simplified in this paper are simple relatively. Moreover, what is worth mentioning is that the result of 20 small limit cycles bifurcating from several fine focuses is good for Z4-equivariant quintic system and the results where multiple singular points become isochronous centers at the same time are less in published references.

关键词： Z4-equivariant focal value center limit cycles isochronous center

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QUALITATIVE THEORY OF THE QUADRATIC DIFFERENTIAL SYSTEMS (II)-ERGODICITY OF LIMIT CYCLES

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Annals of Differential Equations 1998年第2期14卷 294-303页

作者：叶彦谦南京大学

In this paper we study the variation of limit cycles around different foci when a coefficient in the equation of the quadratic differential system varies.

关键词： quadratic differential system anti-saddle limit cycle,weak focus focal value isocline.

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