Sil′nikov Bifurcations in Generic 4-Unfoldings of a Codimension-4 Singularity
✍ Scribed by S. Ibanez; J.A. Rodriguez
- Publisher
- Elsevier Science
- Year
- 1995
- Tongue
- English
- Weight
- 622 KB
- Volume
- 120
- Category
- Article
- ISSN
- 0022-0396
No coin nor oath required. For personal study only.
✦ Synopsis
The main aim of this paper is to prove analytically the existence of homoclinic orbits of focus-saddle type in generic unfoldings of a codimension-4 singularity whose 2 -jet in normal form is given by:
[
x_{2} \frac{\partial}{\partial x_{1}}+x_{3} \frac{\partial}{\partial x_{2}}+\left(a x_{1} x_{2}+b x_{1} x_{3}+c x_{2}^{2}\right) \frac{\partial}{\partial x_{3}} \quad \text { with } \quad a \neq 0
]
Moreover, if the parameters are defined on open sets then these homoclinic orbits satisfy the hypothesis of Sil'nikov's theorem. In this way, we are able to give a method of obtaining families of quadratic vector fields on (\mathbb{R}^{3}) which display strange attractors. 1995 Academic Press, Inc