The limit cycles of a class of discontinuous piecewise differential systems

The determination of the maximum number of limit cycles and their possible positions in the plane is one of the most difficult problems in the qualitative theory of planar differential systems. This problem is related to the second part of the unsolved 16th Hilbert's problem. Due to their applications in modelling many natural phenomena, piecewise differential systems have recently attracted big attention. The upper bound number of limit cycles that a class of differential systems may exhibit is typically very difficult to determine. In this work we extend the second part of the 16th Hilbert's problem to the planar discontinuous piecewise differential systems separated by a straight line and formed by an arbitrary linear centre and an arbitrary cubic uniform isochronous centre. We provide for this class of piecewise differential systems an upper bound on its maximal number of limit cycles, and we prove that such an upper bound is reached.