# American Institute of Mathematical Sciences

November  2008, 7(6): 1415-1428. doi: 10.3934/cpaa.2008.7.1415

## Linearization of smooth planar vector fields around singular points via commuting flows

 1 Departament de Matemàtica, Universitat de Lleida, Avda. Jaume II, 69. 25001. Lleida. 2 Departament de Matemàtica. Universitat de Lleida, Avda. Jaume II, 69. 25001. Lleida, Spain

Received  August 2007 Revised  April 2008 Published  August 2008

In this paper we propose a constructive procedure to get the change of variables that linearizes a smooth planar vector field on $\mathbb C^2$ around an elementary singular point (i.e., a singular point with associated eigenvalues $\lambda, \mu \in \mathbb C$ satisfying $\mu$≠$0$) or a nilpotent singular point from a given commutator. Moreover, it is proved that the near--identity change of variables that linearizes the vector field $\mathcal X = (x+\cdots) \partial_x + (y+\cdots) \partial_y$ is unique and linearizes simultaneously all the centralizers of $\mathcal X$. The method is used in order to obtain the linearization of some extracted examples of the existent literature.
Citation: Isaac A. García, Jaume Giné, Susanna Maza. Linearization of smooth planar vector fields around singular points via commuting flows. Communications on Pure and Applied Analysis, 2008, 7 (6) : 1415-1428. doi: 10.3934/cpaa.2008.7.1415
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