Polynomial Differential Systems in \(\mathbb {R}^3\) Having Invariant Weighted Homogeneous Surfaces

  • Thaís Maria Dalbelo
  • Marcelo MessiasEmail author
  • Alisson C. Reinol


In this paper we give the normal form of all polynomial differential systems in \(\mathbb {R}^3\) having a weighted homogeneous surface \(f=0\) as an invariant algebraic surface and characterize among these systems those having a Darboux invariant constructed uniquely using this invariant surface. Using the obtained results we give some examples of stratified vector fields, when \(f=0\) is a singular surface. We also apply the obtained results to study the Vallis system, which is related to the so-called El Niño atmospheric phenomenon, when it has a cone as an invariant algebraic surface, performing a dynamical analysis of the flow of this system restricted to the invariant cone and providing a stratification for this singular surface.


Polynomial differential systems Darboux theory of integrability Invariant algebraic surfaces Weighted homogeneous surfaces Singular varieties Stratified vector fields Vallis system 



The first author was supported by a scholarship of PNPD/CAPES, developed in the Graduate Program of Applied and Computational Mathematics at the School of Science and Technology, Campus of UNESP at Presidente Prudente, SP, Brazil, and also partially supported by CAPES-Program PVE Grant number 88881.068165/2014-01. The second author was supported by FAPESP Grant number 2013/24541-0, by CNPq Grant number 308159/2015-2, and by CAPES-Program CSF-PVE Grant number 88881.030454/2013. The third author was supported by FAPESP Grant number 2013/26602-7. The authors are grateful to N. Grulha and D. Trotman for their careful reading and suggestions about the stratification part of the paper. The authors also thank the anonymous referee for his valuable comments and suggestions, which help them to improve the results and their presentation in this paper.


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Copyright information

© Sociedade Brasileira de Matemática 2017

Authors and Affiliations

  • Thaís Maria Dalbelo
    • 1
  • Marcelo Messias
    • 1
    Email author
  • Alisson C. Reinol
    • 2
  1. 1.Departamento de Matemática e Computação, Faculdade de Ciências e TecnologiaUNESP-Univ Estadual PaulistaPresidente PrudenteBrazil
  2. 2.Departamento de Matemática, Intituto de Biociências, Letras e Ciências ExatasUNESP-Univ Estadual PaulistaSão José do Rio PretoBrazil

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