Finitude of Limit Cycles of Linear Piecewise ODEs in the Cylinder

Let $x^{\mathrm{\prime }}=S(t,x)$ be a differential equation in the cylinder, linear piecewise in $x$ and with trigonometric coefficients in $t$. In this paper, we provide an upper bound on the number of limit cycles in terms of the number of regions of the piecewise equation and the degree of the coefficients, that is, an analogue of Hilbert’s 16th problem in this context.