On the limit cycles of a quintic planar vector field  

作　　者：Yu-hai WU Li-xin TIAN Mao-an HAN 

机构地区：[1]Department of Mathematics,Jiangsu University,Zhenjiang 212013,China [2]Department of Mathematics,Shanghai Normal University,Shanghai 200234,China

出　　处：《Science China Mathematics》2007年第7期925-940,共16页中国科学：数学（英文版）

基　　金：Supported by the Fund of Youth of Jiangsu University(Grant No.05JDG011);the National Natural Science Foundation of China(Nos.90610031,10671127);the Outstanding Personnel Program in Six Fields of Jiangsu Province(Grant No.6-A-029);Shanghai Shuguang Genzong Project(Grant No.04SGG05)

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 = 52, where H(5) is the Hilbert number for quintic polynomial systems. The results obtained are useful to study the weakened 16th Hilbert problem.This paper concerns the number and distributions of limit cycles in a Z₂-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.

关 键 词：double homoclinic loop Melnikov function STABILITY BIFURCATION limit cycles CONFIGURATION 34C07 34C23 34C37 37G15 

分 类 号：O193[理学—数学]