Bifurcation values for a family of planar vector fields of degree five

TL;DR

This paper investigates the bifurcation and limit cycle behavior of a degree five planar vector field, establishing the existence, uniqueness, and hyperbolicity of a limit cycle within a specific parameter interval using novel algebraic and analytical methods.

Contribution

It introduces new tools based on algebraic curves and double discriminants to determine bifurcation values and limit cycle properties in complex polynomial vector fields.

Findings

Existence of a unique hyperbolic limit cycle for certain parameter values.

Identification of an interval where the bifurcation parameter lies.

Development of a new method using algebraic curves without contact and double discriminants.

Abstract

We study the number of limit cycles and the bifurcation diagram in the Poincar\'{e} 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 $b\in(0,b^*)$ 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.