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These last years an increasing interest appeared for studying the planardiscontinuous piecewise differential systems motivated by the rich applicationsin modelling real phenomena. One of the difficulties for understanding thedynamics of these systems is the study their limit cycles. In this paper westudy the maximum number of crossing limit cycles of some classes of planardiscontinuous piecewise differential systems separated by a straight line, andformed by combinations of linear centers (consequently isochronous) and cubicisochronous centers with homogeneous nonlinearities. For these classes ofplanar discontinuous piecewise differential systems we solved the extension ofthe 16th Hilbert problem, i.e. we provide an upper bound for their maximumnumber of crossing limit cycles.

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