Poincaré Compactification for Non-polynomial Vector Fields

• Bravo, José Luis ^[1] ; Fernández, Manuel ^[1] ; Teruel, Antonio E ^[2]
  1. [1] Universidad de Extremadura

     Universidad de Extremadura

     Badajoz, España

     
  2. [2] Universitat de les Illes Balears

     Universitat de les Illes Balears

     Palma de Mallorca, España

     
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 19, Nº 1, 2020
• Idioma: inglés
• DOI: 10.1007/s12346-020-00386-1
• Enlaces
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• Resumen

  • In this work a theorical framework to apply the Poincaré compactification technique to locally Lipschitz continuous vector fields is developed. It is proved that these vectors fields are compactifiable in the n-dimensional sphere, though the compactified vector field can be identically null in the equator. Moreover, for a fixed projection to the hemisphere, all the compactifications of a vector field, which are not identically null on the equator are equivalent. Also, the conditions determining the invariance of the equator for the compactified vector field are obtained. Up to the knowledge of the authors, this is the first time that the Poincaré compactification of locally Lipschitz continuous vector fields is studied. These results are illustrated applying them to some families of vector fields, like polynomial vector fields, vector fields defined as a sum of homogeneous functions and vector fields defined by piecewise linear functions.

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