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March, 2004 A characterization of isochronous centres in terms of symmetries
Emilio Freire, Armengol Gasull, Antoni Guillamon
Rev. Mat. Iberoamericana 20(1): 205-222 (March, 2004).


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.


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Emilio Freire. Armengol Gasull. Antoni Guillamon. "A characterization of isochronous centres in terms of symmetries." Rev. Mat. Iberoamericana 20 (1) 205 - 222, March, 2004.


Published: March, 2004
First available in Project Euclid: 2 April 2004

zbMATH: 1087.34010
MathSciNet: MR2076778

Primary: 17B80 , 34C14 , 37C27

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

Rights: Copyright © 2004 Departamento de Matemáticas, Universidad Autónoma de Madrid

Vol.20 • No. 1 • March, 2004
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