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Differential Equations Defined by the Sum of two Quasi-Homogeneous Vector Fields

Published online by Cambridge University Press:  20 November 2018

B. Coll ,
A. Gasull  and
R. Prohens
B. Coll
Affiliation:
Departament de Matemàtiques i Informàtica, Universitat de les Illes Balears, Carretera de Valldemossa, Km7.5, 07071 Palma de Mallorca, Spain
A. Gasull
Affiliation:
Departament de Matemàtiques, Edifici Cc, Universitat Autónoma de Barcelona, 08193 Bellaterra, Barcelona, Spain e-mail: GASULL@mat.uab.es
R. Prohens
Affiliation:
Departament de Matemàtiques, Edifici Cc, Universitat Autónoma de Barcelona, 08193 Bellaterra, Barcelona, Spain
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Abstract

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In this paper we prove, that under certain hypotheses, the planar differential equation: ˙x = X1(x, y) + X2(x, y), ˙y = Y1(x, y) + Y2(x, y), where (Xi, Yi), i = 1, 2, are quasi-homogeneous vector fields, has at most two limit cycles. The main tools used in the proof are the generalized polar coordinates, introduced by Lyapunov to study the stability of degenerate critical points, and the analysis of the derivatives of the Poincar´e return map. Our results generalize those obtained for polynomial systems with homogeneous non-linearities.

Keywords

34C0558F21
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 49 , Issue 2 , 01 April 1997 , pp. 212 - 231
Copyright
Copyright © Canadian Mathematical Society 1997

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