Normal Form for a Class of Three-Dimensional Systems with Free-Divergence Principal Part
We present the basic ideas of the Normal Form Theory by using quasi-homogeneous expansions of the vector field, where the structure of the normal form is determined by the principal part of the vector field. We focus on a class of tridimensional systems whose principal part is the coupling of a Hamiltonian planar system and an unidimensional system, in such a way that the quoted principal part does not depend on the last variable and has free divergence. Our study is based on several decompositions of quasi-homogeneous vector fields. An application, corresponding to the coupling of a Takens-Bogdanov and a saddle-node singularities, (in fact, it is a triple-zero singularity with geometric multiplicity two), that falls into the class considered, is analyzed.
KeywordsNormal forms Conservative-disipative splitting Hamiltonian Homological operator Lie operator Quasi-homogeneous
This work has been partially supported by Ministerio de Ciencia y Tecnología, Plan Nacional I+D+I co-financed with FEDER funds, in the frame of the project MTM2014-56272-C2-02, and by Consejería de Educación y Ciencia de la Junta de Andalucía (FQM-276 and P12-FQM-1658).
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