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arXiv · 1710.00883

Generic 2-parameter perturbations of parabolic singular points of vector fields in C

Martin Klimes·Christiane Rousseau
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Abstract

We describe the equivalence classes of germs of generic $2$-parameter families of complex vector fields $\dot z = \omega_\epsilon(z)$ on $\mathbb{C}$ unfolding a singular parabolic point of multiplicity $k+1$: $\omega_0= z^{k+1} +o(z^{k+1})$. The equivalence is under conjugacy by holomorphic change of coordinate and parameter. As a preparatory step, we present the bifurcation diagram of the family of vector fields $\dot z = z^{k+1} + \epsilon_1 z + \epsilon_0$ over $\mathbb{CP}^1$. This presentation is done using the new tools of periodgon and star domain. We then provide a description of the modulus space and (almost) unique normal forms for the equivalence classes of germs.

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Martin Klimes, Christiane Rousseau. 2017-10-02. Generic 2-parameter perturbations of parabolic singular points of vector fields in C. https://arxiv.org/abs/1710.00883

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