## Generic 3-parameter families of vector fields on the plane, unfolding a singularity with nilpotent linear part. The cusp case of codimension 3.(English)Zbl 0608.58034

A cusp type germ of vector fields is a $${\mathcal C}^{\infty}$$ germ at $$0\in {\mathbb{R}}^ 2$$, whose 2-jet is $${\mathcal C}^{\infty}$$ conjugate to $$y\frac{\partial}{\partial x}+(\alpha x^ 2+\beta xy)\frac{\partial}{\partial y}$$ with $$\alpha\neq 0$$. We define a submanifold of codimension 5 in the space of germs $$\Sigma ^ 3_{C\pm}$$, consisting of germs of cusp type whose 4-jet is $${\mathcal C}^ 0$$ equivalent to $$y\frac{\partial}{\partial x}+(x^ 2\pm x^ 3y)\frac{\partial}{\partial y}$$. Our main result can be stated as follows: any local 3-parameter family in $$(0,0)\in {\mathbb{R}}^ 2\times {\mathbb{R}}^ 3$$, cutting $$\Sigma ^ 3_{C\pm}$$ transversally in (0,0) is fibre-$${\mathcal C}^ 0$$ equivalent to $\tilde X^{\pm}_{\lambda} = y\frac{\partial}{\partial x}+(x^ 2+\mu +y(\nu _ 0+\nu _ 1x\pm x^ 3))\frac{\partial}{\partial y}.$

### MSC:

 37G99 Local and nonlocal bifurcation theory for dynamical systems 37G15 Bifurcations of limit cycles and periodic orbits in dynamical systems

### Keywords:

bifurcations; planar vector fields; singularity; limit cycles
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### References:

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