A characterization of isochronous centres in terms of symmetries.
Summary: 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.
scientific article; zbMATH DE number 2104144