Systems bifurcation of limit cycles for a family of discontinuous piecewise isochronous differential systems separated by irregular line

Abstract

The study of Hilbert’s 16th problem for piecewise linear differential systems has received significant attention from many researchers. It was shown that the upper bound for the maximum number of limit cycles can vary according to the configuration of the discontinuous curve. The family of discontinuous piecewise differential systems formed by linear isochronous centers or four families of quadratic isochronous centers separated by a straight line have been studied, and the authors have found at most two limit cycles. In this paper, we study the same family but instead of a straight line, we consider an irregular line separation, and we prove that there are at most five crossing limit cycles intersecting the separation curve at two points.