DIFFERENTIAL-EQUATIONS DEFINED BY THE SUM OF 2 QUASI-HOMOGENEOUS VECTOR-FIELDS

In this paper we prove, that under certain hypotheses, the planar diff erential equation: x = X-i (x,y) + X-2(x,y), y = Y-1(x,y) + Y-2(x,y), where (X-i, Y-i), i = 1, 2, are quasi-homogeneous vector fields, has a t most two limit cycles. The main tools used in the proof are the gene ralized polar coordinates, introduced by Lyapunov to study the stabili ty of degenerate critical points, and the analysis of the derivatives of the Poincare return map. Our results generalize those obtained for polynomial systems with homogeneous non-linearities.