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

Dumortier, F.; Roussarie, R.; Sotomayor, J.

Ergodic Theory Dyn. Syst. 7, 375-413 (1987).

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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