Normal Form for a Class of Three-Dimensional Systems with Free-Divergence Principal Part

  • Antonio Algaba
  • Natalia Fuentes
  • Estanislao GameroEmail author
  • Cristóbal García
Part of the Understanding Complex Systems book series (UCS)


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.


Normal 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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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Antonio Algaba
    • 1
  • Natalia Fuentes
    • 1
  • Estanislao Gamero
    • 2
    Email author
  • Cristóbal García
    • 1
  1. 1.Department of Integrated SciencesInvestigation Center of Theoretical Physics and Mathematic FIMAT, Huelva UniversityHuelvaSpain
  2. 2.Department of Applied Mathematic IIE.T.S.I. Sevilla UniversitySevillaSpain

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