Differential equations defined by the sum of two quasi-homogeneous vector fields

B. Coll, A. Gasull, R. Prohens

Research output: Contribution to journalArticleResearchpeer-review

28 Citations (Scopus)

Abstract

In this paper we prove, that under certain hypotheses, the planar differential equation: ẋ = X1(x, y) + X2(x, y), ẏ = 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 Poincaré return map. Our results generalize those obtained for polynomial systems with homogeneous non-linearities.
Original languageEnglish
Pages (from-to)212-231
JournalCanadian Journal of Mathematics
Volume49
Issue number2
DOIs
Publication statusPublished - 1 Jan 1997

Fingerprint Dive into the research topics of 'Differential equations defined by the sum of two quasi-homogeneous vector fields'. Together they form a unique fingerprint.

Cite this