An extension of the 16th Hilbert problem for continuous piecewise linear-quadratic centers separated by a non-regular line

In the last few decades, there has been much interest in studying piecewise differential systems. This is mainly due to the fact that these differential systems allow us to modelize many natural phenomena. In order to describe the dynamics of a differential system, we need to control its periodic orbits and, especially, its limit cycles. In particular, providing an upper bound for the maximum number of limit cycles that such differential systems can exhibit would be desirable, that is solving the extended 16th Hilbert problem. In general, this is an unsolved problem. In this paper, we give an upper bound for the maximum number of limit cycles that a class of continuous piecewise differential systems formed by an arbitrary linear center and an arbitrary quadratic center separated by a non-regular line can exhibit. So for this class of continuous piecewise differential systems, we have solved the extended 16th Hilbert problem, and the upper bound found is seven. The question whether this upper bound is sharp remains open.