Limit cycles of discontinuous piecewise differential systems formed by linear and cubic isochronous centers

Abstract

One of the main problems in the qualitative theory of the planar differential systems is to control the existence and the number of their limit cycles. There are many researchers who tried to solve this problem for special classes of planar differential systems, see for instance [7, 16]. In this paper, we study the maximum number of limit cycles for discontinuous planar piecewise differential systems formed by four classes of isochronous cubic centers separated by irregular straight line. We provide a sharp upper bound for the maximum number of crossing limit cycles that these classes of discontinuous piecewise differential systems can exhibit. Therefore, we will solve the extended of the 16th Hilbert problem for these classes. Keywords: limit cycles; cubic isochronous centers; linear differential center.