Bifurcation of critical periods at the outer boundary for some potential vector fields [ Back ]

15:30 to 16:30
UAB - Dept. Matemàtiques (C1/-128)
David Rojas


Consider a planar family of planar differential systems with a center at $p$. The period function assigns to each periodic orbit in the period annulus its period. The problem of bifurcation of critical periods have been studied and there are three different situations to consider: bifurcations from the center, bifurcations from the interior of the period annulus and bifurcation from the outer boundary of the period annulus. The bifurcation of critical periods from the inner boundary is completely understood thanks to C. Chicone and M. Jacobs. We study the bifurcation of critical periods from the outer boundary, which has an added difficulty since the period function can not be analytically extended. We focus the study in the case of families of potential vector fields such that the outer boundary is reached with infinite energy level. We shall present a result concerning the regularity of a certain value of the parameter space and another result concerning the criticality of a bifurcation value of the parameter space. Particularly, we found the bifurcation curves associated to the outer boundary of the bi-parametric family of potential systems $X_{p,q}=-ydx+V_{p,q}(x)dy$ with $V_{p,q}'(x)=(x+1)^p-(x+1)^q$, for $q+1<0$.