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| HOMOCLINIC BIFURCATION WITH CODIMENSION 3 |
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Citation： |
Zhu Deming.HOMOCLINIC BIFURCATION WITH CODIMENSION 3[J].Chinese Annals of Mathematics B,1994,15(2):205~216 |
| Page view： 1088
Net amount： 691 |
Authors： |
Zhu Deming; |
Foundation： |
Project supported by the National Natural Science Foundation of China |
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| Abstract： |
First it is proved that both the integral of the divergence and the
Melnikov function are invariants of the $C^2$ transformation. Then, the
problem of the planar homoclinic bifurcation with codimension $3$ is
considered. It is proved that, in a small neighborhood of the origin in
the parameter space of a $C^r$ ($r \ge 5$) system, there exist exactly
two $C^{r-1}$ semi-stable-limit-cycle branching surfaces, and their
common boundary is a unique $C^{r-1}$ three-multiple-limit-cycle
branching curve. The bifurcation pictures and the asymptotic expansions
of the bifurcation functions are given. The stability criterion for the
homoclinic loop is also obtained when the integral of the divergence is
zero. The proof of the auxiliary theorems will be presented in [16].
\endabstract |
Keywords： |
Homoclinic bifurcation, Codimension,Semi-stable-limit-cycle branch,Three-multiple-limit-cycle branch. |
Classification： |
34C05, 34C37 |
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