| On the limit cycles of a quintic planar vector field |
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| 作者姓名： | Yu-hai WU~ |
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| 作者单位： | Yu-hai WU（Department of Mathematics, Jiangsu University, Zhenjiang 212013, China） ;
Li-xin TIAN（Department of Mathematics, Jiangsu University, Zhenjiang 212013, China） ;
Mao-an HAN（Department of Mathematics, Shanghai Normal University, Shanghai 200234, China） ; |
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| 基金项目： | 江苏大学校科研和教改项目;国家自然科学基金;江苏省六大人才高峰基金;上海市教委"曙光计划" |
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| 摘 要： | This paper concerns the number and distributions of limit cycles in a Z_2-equivariant quintic planar vector field.25 limit cycles are found in this special planar polynomial system and four different configurations of these limit cycles are also given by using the methods of the bifurcation theory and the qualitative analysis of the differential equation.It can be concluded that H(5)≥25=5~2, where H(5)is the Hilbert number for quintic polynomial systems.The results obtained are useful to study the weakened 16th Hilbert problem.
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| 收稿时间： | 3 August 2006 |
| 修稿时间： | 8 January 2007 |
On the limit cycles of a quintic planar vector field |
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| Yu-hai WU.On the limit cycles of a quintic planar vector field[J].Science in China(Mathematics),2007,50(7):925-940. |
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| Authors: | Yu-hai Wu Li-xin Tian Mao-an Han |
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| Institution: | 1. Department of Mathematics, Jiangsu University, Zhenjiang 212013, China
2. Department of Mathematics, Shanghai Normal University, Shanghai 200234, China |
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| Abstract: | This paper concerns the number and distributions of limit cycles in a Z
2-equivariant quintic planar vector field. 25 limit cycles are found in this special planar polynomial system and four different
configurations of these limit cycles are also given by using the methods of the bifurcation theory and the qualitative analysis
of the differential equation. It can be concluded that H(5) ⩾ 25 = 52, where H(5) is the Hilbert number for quintic polynomial systems. The results obtained are useful to study the weakened 16th Hilbert
problem.
Supported by the Fund of Youth of Jiangsu University (Grant No. 05JDG011), the National Natural Science Foundation of China
(Nos. 90610031, 10671127), the Outstandign Personnel Program in Six Fields of Jiangsu Province (Grant No. 6-A-029) and Shanghai
Shuguang Genzong Project (Grant No. 04SGG05) |
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| Keywords: | double homoclinic loop Melnikov function stability bifurcation limit cycles configuration |
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