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Every Period Annulus is Both Reversible and Symmetric

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Abstract

We prove that for every planar differential system with a period annulus there exists a unique involution \(\sigma \) such that the system is \(\sigma \)-symmetric. We also prove that, given a system with a period annulus and a global section \(\delta \), there exist a unique involution \(\sigma \) such that the system is \(\sigma \)-reversible and \(\delta \) is the fixed points curve of \(\sigma \). As a consequence, every system with a period annulus admits infinitely many reversibilities.

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Authors and Affiliations

  1. Dipartimento di Matematica, Università di Trento, 38123, Povo, TN, Italy

    Marco Sabatini

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  1. Marco Sabatini
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Correspondence to Marco Sabatini.

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This paper was partially supported by the GNAMPA, Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni. The results of this paper were presented at the meeting AQTDE2015, held in Tarragona, April 20–24, 2015. A preliminary version can be found at http://arxiv.org/abs/1504.04530.

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Sabatini, M. Every Period Annulus is Both Reversible and Symmetric. Qual. Theory Dyn. Syst. 16, 175–185 (2017). https://doi.org/10.1007/s12346-015-0183-7

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  • DOI: https://doi.org/10.1007/s12346-015-0183-7

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