The period function for some planar piecewise vector fields

Abstract

In this work, we study the period function for fixed families of piecewise differential vector fields with a line of discontinuities. These systems, indistinctly called piecewise or nonsmooth, appear in several applications, including among others optimal control, nonsmooth mechanics, and robotic manipulation. For one family, by using a method based upon Picard-Fuchs equations for algebraic curves, we characterize the global behavior of the period function. That is, we determine regions in the parameter space for which the corresponding period function is monotonous or it has critical periods. Furthermore, in one of these families we study the bifurcation of critical periods in the interior of the period annulus from the weak center and from the isochronous center by using the calculation of the Taylor developments of the periods constants near the center. We further present the beginning of the study of the global behavior of the period function for the planar piecewise linear system that contains a period annulus at infinity.