Asymptotic phase and invariant foliations near periodic orbits

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 r...

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Bibliographic Details
Published in:Proceedings of the American Mathematical Society 2006-10, Vol.134 (10), p.2989-2996
Main Author: Dumortier, Freddy
Format: Article
Language:English
Subjects:
Coordinate systems
Electrical phases
Exact sciences and technology
Mathematical analysis
Mathematical theorems
Mathematics
Natural numbers
Ordinary differential equations
Periodic orbits
Riemann manifold
Sciences and techniques of general use
Vector fields
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Summary: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.
ISSN:0002-9939
1088-6826
DOI:10.1090/S0002-9939-06-08392-4