A characterization of isochronous centres in terms of symmetries

Armengol Gasull, Emilio Freire, Antoni Guillamon

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18 Citations (Scopus)


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.
Original languageEnglish
Pages (from-to)205-222
JournalRevista Matematica Iberoamericana
Issue number1
Publication statusPublished - 1 Jan 2004


  • Groups of symmetries
  • Isochronous centres
  • Normalizers
  • Quadratic-like Hamiltonian systems


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