Limit cycles of piecewise differential equations on the cylinder

We consider the discontinuous piecewise differential equations of the form [Formula presented] where a0(t),a1(t),...,an(t) and b0(t),b1(t),...,bm(t) are 2π-periodic functions in the variable t, and we study the number of limit cycles of such equations on the cylinder. In this way we give exact bounds for the maximum number of limit cycles that the piecewise differential equations have in function of n and m. Note that usually the discontinuous piecewise differential systems are discontinuous in the dependent variable, here the system is discontinuous in the independent variable