Homoclinic bifurcations in symmetric unfoldings of a singularity with three-fold zero eigenvalue. (English) Zbl 1095.34024
The degenerate singularities and their unfoldings of plane vector fields are well studied. As for singularities in vector fields of dimension more than two much less results have been obtained due to the existence of global bifurcations, such as homoclinic or heteroclinic orbits, and chaotic behavior.
The paper studies a singularity at the origin with three-fold zero eigenvalue for symmetric vector fields with nilpotent linear part and 3-jet \(C^{\infty}\)-equivalent to
\[ y\,\frac{\partial}{\partial x}+z\,\frac{\partial}{\partial y}+ ax^2y\,\frac{\partial}{\partial z} \] with \(a\neq 0\). Under some conditions, several subfamilies of the symmetric versal unfoldings of this singularity are obtained by using normal form and blow-up methods. The local and global bifurcation behavior is derived. For some subfamilies of the symmetric versal unfoldings of this singularity, the existence of the Sil’nikov homoclinic bifurcation is proved by applying the generalized Mel’nikov methods of a homoclinic orbit to a hyperbolic or nonhyperbolic equilibrium.
The paper studies a singularity at the origin with three-fold zero eigenvalue for symmetric vector fields with nilpotent linear part and 3-jet \(C^{\infty}\)-equivalent to
\[ y\,\frac{\partial}{\partial x}+z\,\frac{\partial}{\partial y}+ ax^2y\,\frac{\partial}{\partial z} \] with \(a\neq 0\). Under some conditions, several subfamilies of the symmetric versal unfoldings of this singularity are obtained by using normal form and blow-up methods. The local and global bifurcation behavior is derived. For some subfamilies of the symmetric versal unfoldings of this singularity, the existence of the Sil’nikov homoclinic bifurcation is proved by applying the generalized Mel’nikov methods of a homoclinic orbit to a hyperbolic or nonhyperbolic equilibrium.
Reviewer: Eugene Ershov (St. Petersburg)
MSC:
| 34C37 | Homoclinic and heteroclinic solutions to ordinary differential equations |
| 34C05 | Topological structure of integral curves, singular points, limit cycles of ordinary differential equations |
| 34C20 | Transformation and reduction of ordinary differential equations and systems, normal forms |
| 34C23 | Bifurcation theory for ordinary differential equations |
| 37G05 | Normal forms for dynamical systems |
| 37G20 | Hyperbolic singular points with homoclinic trajectories in dynamical systems |
Keywords:
singularity; symmetric unfolding; homoclinic orbit, Sil’nikov bifurcation; normal form; blow-up; generalized Mel’nikov methodsReferences:
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