||We study the number of limit cycles and the bifurcation diagram in the Poincar´ sphere of a one-parameter family of planar diﬀerential equations of degree e ˙ ﬁve x = Xb (x) which has been already considered in previous papers. We prove that there is a value b∗ > 0 such that the limit cycle exists only when b ∈ (0, b∗ ) and that it is unique and hyperbolic by using a rational Dulac function. Moreover we provide an interval of length 27/1000 where b∗ lies. As far as we know the tools used to determine this interval are new and are based on the construction of algebraic curves without contact for the ﬂow of the diﬀerential equation. These curves are obtained using analytic information about the separatrices of the inﬁnite critical points of the vector ﬁeld. To prove that the Bendixson-Dulac Theorem works we develop a method for studying whether one-parameter families of polynomials in two variables do not vanish based on the computation of the so called double discriminant.
||Número d'acord de subvenció MCYT/MTM2008-03437|
||Número d'acord de subvenció CIRIT/2009/SGR-410
||Tots els drets reservats.
||article ; recerca ; preprint
Double discriminant ;
Dulac function ;
Phase portrait on the Poincaré sphere ;
Polynomial planar system ;
Uniqueness of limit cycles
||Discrete and Continuous Dynamical Systems. Series A, Vol. 35 Núm. 2 (2015) , p. 669-701, ISSN 1553-5231