Abstract
We analyze the isochronicity of centres, by means of a procedure to obtain hypernormal forms (simplest normal forms) for the Hopf bifurcation, that uses the theory of transformations based on the Lie transforms. We establish the relation between the period constants and the normal form coefficients, and prove that an equilibrium point is an isochronous centre if and only if a property of commutation holds. Also, we give necessary and sufficient conditions, expressed in terms of the Lie product, to determine if an equilibrium point is a centre. Several examples are also included in order to show the usefulness of the method. In particular, the isochronicity of the origin for the Lienard equation is analyzed in some cases.
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Algaba, A., Freire, E. & Gamero, E. Isochronicity via normal form. Qual. Th. Dyn. Syst 1, 133–156 (2000). https://doi.org/10.1007/BF02969475
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DOI: https://doi.org/10.1007/BF02969475