Crossing limit cycles of planar discontinuous piecewise differential systems formed by isochronous centres

These last years an increasing interest appeared in studying the planar discontinuous piecewise differential systems motivated by the rich applications in modelling real phenomena. The understanding of the dynamics of these systems has many difficulties. One of them is the study of their limit cycles. In this paper, we study the maximum number of crossing limit cycles of some classes of planar discontinuous piecewise differential systems separated by a straight line and formed by combinations of linear centres (consequently isochronous) and cubic isochronous centres with homogeneous nonlinearities. For these classes of planar discontinuous piecewise differential systems we solved the extension of the 16th Hilbert problem, i.e. we provide an upper bound for their maximum number of crossing limit cycles.