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