On the non-existence of isochronous centers in planar discontinuous differential systems

The determination of whether a center is isochronous or not is a well-known problem in the qualitative theory of planar systems. In this paper, we explore planar piecewise discontinuous differential systems characterized by a straight switching line $y=0$ and $x=0$, respectively. Our investigation reveals that such systems do not possess an isochronous center at the origin.