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

Open Access article
 Printed: Apr 1997
  • B. Coll
  • A. Gasull
  • R. Prohens
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In this paper we prove, that under certain hypotheses, the planar differential equation: $\dot x=X_1(x,y)+X_2(x,y)$, $\dot y=Y_1(x,y)+Y_2(x,y)$, where $(X_i,Y_i)$, $i=1$, $2$, 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.
MSC Classifications: 34C05, 58F21 show english descriptions Location of integral curves, singular points, limit cycles
unknown classification 58F21
34C05 - Location of integral curves, singular points, limit cycles
58F21 - unknown classification 58F21

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