| Homoclinic Bifurcations in Symmetric Unfoldings of a Singularity with Three-fold Zero Eigenvalue |
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| 作者姓名： | JianHuaSUN |
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| 作者单位： | DepartmentofMathematics,NanjingUniversity,Nanjing210008,P.R.China |
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| 摘 要： | In this paper we study the singularity at the origin with three-fold zero eigenvalue forsymmetric vector fields with nilpotent linear part and 3-jet C^∞-equivalent to y δ/δx zδ/δy ax^2yδ/δz with a≠0. We first obtain several subfamilies of the symmetric versal unfoldings of this singularityby using the normal form and blow-up methods under some conditions, and derive the local and global bifurcation behavior, then prove analytically the existence of the Sil‘nikov homoclinic bifurcation for some subfamilies of the symmetric versal unfoldings of this singularity, by using the generalized Mel‘nikov methods of a homoclinic orbit to a hyperbolic or non-hyperbolic equilibrium in a highdimensional space.
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| 关 键 词： | 奇异性 对称展开 同宿轨 零特征值 广义Mel＇nikov法 微分方程 |
| 收稿时间： | 21 September 2001 |
Homoclinic Bifurcations in Symmetric Unfoldings of a Singularity with Three–fold Zero Eigenvalue |
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| JianHuaSUN.Homoclinic Bifurcations in Symmetric Unfoldings of a Singularity with Three-fold Zero Eigenvalue[J].Acta Mathematica Sinica,2005,21(1):65-80. |
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| Authors: | Jian Hua Sun |
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| Institution: | (1) Department of Mathematics, Nanjing University, Nanjing 210008, P. R. China |
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| Abstract: | In this paper we study the singularity at the origin with three–fold zero eigenvalue for
symmetric vector fields with nilpotent linear part and 3–jet C∞–equivalent to
with a ≠ 0. We first obtain several subfamilies of the symmetric versal unfoldings of this singularity
by using the normal form and blow–up methods under some conditions, and derive the local and
global bifurcation behavior, then prove analytically the existence of the Sil'nikov homoclinic bifurcation
for some subfamilies of the symmetric versal unfoldings of this singularity, by using the generalized
Mel'nikov methods of a homoclinic orbit to a hyperbolic or non–hyperbolic equilibrium in a highdimensional
space.
Project supported by the National Natural Science Foundation of China (No. 10171044) and the Foundation
for University Key Teachers of the Ministry of Education |
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| Keywords: | Singularity Symmetric unfolding Homoclinic orbit Sil’ nikov bifurcation Normal form Blow– up Generalized Mel'nikov methods |
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