A characterization of isochronous centres in terms of symmetries

• Autores: Armengol Gasull i Embid , Antoni Guillamon Grabolosa , Emilio Freire Macías
• Localización: Revista matemática iberoamericana, ISSN 0213-2230, Vol. 20, Nº 1, 2004, págs. 205-222
• Idioma: inglés
• DOI: 10.4171/rmi/386
• Títulos paralelos:
  • Una caracterización de los centros isócronos en términos de simetrías.
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• Resumen

  • We present a description of isochronous centres of planar vector fields X by means of their groups of symmetries. More precisely, given a normalizer U of X (i.e., [X,U]= µ X, where µ is a scalar function), we provide a necessary and sufficient isochronicity condition based on µ. This criterion extends the result of Sabatini and Villarini that establishes the equivalence between isochronicity and the existence of commutators ([X,U]= 0). We put also special emphasis on the mechanical aspects of isochronicity; this point of view forces a deeper insight into the potential and quadratic-like Hamiltonian systems. For these families we provide new ways to find isochronous centres, alternative to those already known from the literature.