On Hilbert’s 16th Problem for Some Discontinuous Piecewise Differential Systems

Abstract

In this paper, we study the number of limit cycles for discontinuous piecewise differential systems formed by two differential systems separated by a straight line, when these differential systems are quadratic isochronous centers in a half-plane and the cubic isochronous centers in the other one. We deal with eight classes of discontinuous piecewise differential systems formed by these types of isochronous centers, and we provide an upper bound for the maximum number of limit cycles for these classes that can exhibit. Also, we show that the corresponding maximum number of limit cycles in four classes is reached.