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Characterization of isochronous foci for planar analytic differential systems

Published online by Cambridge University Press:  12 July 2007

Jaume Giné  and
Maite Grau
Jaume Giné
Affiliation:
Departament de Matemática, Universitat de Lleida, Avda. Jaume II 69, 25001 Lleida, Spain. ([email protected])
Maite Grau
Affiliation:
Departament de Matemática, Universitat de Lleida, Avda. Jaume II 69, 25001 Lleida, Spain. ([email protected])
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Abstract

We consider the two-dimensional autonomous systems of differential equations of the form where P(x,y) and Q(x,y) are analytic functions of order greater than or equal to 2. These systems have a focus at the origin if λ ≠ 0, and have either a centre or a weak focus if λ = 0. In this work we study the necessary and sufficient conditions for the existence of an isochronous critical point at the origin. Our result is, to the best of our knowledge, original when applied to weak foci and gives known results when applied to strong foci or to centres.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 135 , Issue 5 , October 2005 , pp. 985 - 998
Copyright
Copyright © Royal Society of Edinburgh 2005

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