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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Asymptotic phase and invariant foliations near periodic orbits
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by Freddy Dumortier PDF
Proc. Amer. Math. Soc. 134 (2006), 2989-2996 Request permission

Abstract:

The paper deals with asymptotic phase and invariant foliations near periodic orbits, extending for two-dimensional smooth vector fields results that have been obtained by Chicone and Liu (2004). The problem of the existence of asymptotic phase is completely solved for analytic vector fields and is reduced to a problem of infinite codimension for $C^{\infty }$ systems. Moreover it is proven that whenever asymptotic phase occurs, or in other words, when the periodic orbit is isochronous, then there also exists a $C^{\infty }$ foliation, with leaves transversally cutting the periodic orbit and invariant under the flow of the vector field. The paper also provides some results in three dimensions.
References
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Additional Information
  • Freddy Dumortier
  • Affiliation: Universiteit Hasselt, Campus Diepenbeek, Agoralaan - Gebouw D, B-3590 Diepenbeek, Belgium
  • Email: freddy.dumortier@uhasselt.be
  • Received by editor(s): May 4, 2005
  • Published electronically: May 5, 2006
  • Additional Notes: The author thanks the Ministerio de Educacion y Ciencia from Spain for their financial support and the Centre de Recerca Matemàtica at Bellaterra for their hospitality during the preparation of this paper.
  • Communicated by: Carmen C. Chicone
  • © Copyright 2006 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 134 (2006), 2989-2996
  • MSC (2000): Primary 34D05, 34C07, 34C20
  • DOI: https://doi.org/10.1090/S0002-9939-06-08392-4
  • MathSciNet review: 2231624