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A characterization of isochronous centres in terms of symmetries. (English) Zbl 1087.34010

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.

MSC:

34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
34C14 Symmetries, invariants of ordinary differential equations
37J45 Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods (MSC2010)
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References:

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