Bifurcation of zeros in translated families of functions and applications [ Back ]

UAB - Dept. Matemàtiques (C1/-128)
Jordi Villadelprat
Universitat Rovira i Vergili


This talk deals with the study of bifurcation of critical periodic orbits and limit cycles of a family of planar vector fields in a neighbourhood of a polycycle. A key problem in these studies is the breaking of separatrices of the polycycle. The case of cyclicity of hyperbolic polycycles has been extensively studied by Roussarie, Mourtada, Il’yashenko and many others. On the other hand, in our study of the period function of the Loud’s centers, we gave a conjecture for the bifurcation of critical periodic orbits. We could not prove it in full generality due in part to some phenomena of breaking of separatrices of the polycycle bounding the period annulus. Our aim is to tackle the simplest setting of breaking one separatrix (or two separatrices in presence of symmetry). Both problems lead to the following type of equation
where $s=0$ corresponds to the polycycle, $s\geq 0$ parametrizes the monodromic region, $\varepsilon\approx 0$ is the parameter controlling the breaking of the separatrix and $\mu$ are the non-essential parameters of the family.

The results that I am going to explain are a joint work, still in progress, with D. Marin (Universitat Autonoma de Barcelona) and P. Mardesic (Université de Bourgogne).