Abstract:

We study the number of limit cycles and the bifurcation diagram in the Poincaré sphere of a one-parameter family of  planar differential equations of degree five $dot {bf x}=X_b({bf x})$ 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 $bin(0,b^*)$ 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. The uniqueness of the limit cycle  follows by applying the Bendixson-Dulac Theorem.

This talk is based on a joint paper with A. Gasull and H. Giacomini.