Algebraic and analytic tools to study critical periodic orbits [ Back ]

UAB - Dept. Matemàtiques (C1/-128)
Jordi Villadelprat


In this talk I will present some recent results concerning critical period orbits, which in the context of the period function play an equivalent role to the limit cycles. We consider analytic planar differential systems having a first integral of the form $H(x,y)=A(x)+B(x)y+C(x)y^2$ and an integrating factor $\kappa(x)$ not depending on $y.$
We will discuss three results. The first one provides a characterization of isochronicity, a criterion to bound the number of critical periodic orbits and a necessary condition for the period function to be monotone.
The second result enables to bound the multiplicity of a function defined implicitly in terms of the multiplicity of an appropriate resultant. This second result is intended for being applied in combination with the first one in an algebraic setting that we shall specify. Finally, the third result is devoted to study the number of critical periodic orbits bifurcating from the period annulus of an isochrone perturbed linearly inside a family of centers.

The talk is based on a joint paper with Antonio Garijo (URV).