Revista Matemática Iberoamericana

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Volume 20, Issue 1, 2004, pp. 205–222
DOI: 10.4171/RMI/386

Published online: 2004-04-30

A characterization of isochronous centres in terms of symmetries

Emilio Freire[1], Armengol Gasull[2] and Antoni Guillamon[3]

(1) Universidad de Sevilla, Spain
(2) Universitat Autònoma de Barcelona, Bellaterra, Spain
(3) Universitat Politècnica de Catalunya, Barcelona, Spain

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., $[X,U]=\mu X$, where $\mu$ is a scalar function), we provide a necessary and sufficient isochronicity condition based on $\mu$. 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.

Keywords: Isochronous centres, quadratic-like Hamiltonian systems, groups of symmetries, normalizers

Freire Emilio, Gasull Armengol, Guillamon Antoni: A characterization of isochronous centres in terms of symmetries. Rev. Mat. Iberoam. 20 (2004), 205-222. doi: 10.4171/RMI/386