## Generic $2$-parameter perturbations of parabolic singular points of vector fields in $\mathbb {C}$

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- by Martin Klimeš and Christiane Rousseau PDF
- Conform. Geom. Dyn.
**22**(2018), 141-184 Request permission

## 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 $\dot z = z^{k+1}+\epsilon _1z+\epsilon _0$ over $\mathbb {C}\mathbb {P}^1$. 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.## References

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## Additional Information

**Martin Klimeš**- Affiliation: Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria
- Email: martin.klimes@univie.ac.at
**Christiane Rousseau**- Affiliation: Département de mathématiques et de statistique, Université de Montréal, C.P. 6128, Succursale Centre-ville, Montréal Quebec, H3C 3J7, Canada
- MR Author ID: 192734
- Email: rousseac@dms.umontreal.ca
- Received by editor(s): October 3, 2017
- Received by editor(s) in revised form: July 24, 2018
- Published electronically: September 7, 2018
- Additional Notes: The first author thanks CRM where this research was first initiated.

The second author was supported by NSERC in Canada. - © Copyright 2018 American Mathematical Society
- Journal: Conform. Geom. Dyn.
**22**(2018), 141-184 - MSC (2010): Primary 37F75, 32M25, 32S65, 34M99
- DOI: https://doi.org/10.1090/ecgd/325
- MathSciNet review: 3851392