On stability of periodic solutions in non-homogeneous Hill´s equation

In this note we study the stability property of periodic solutions of Hill´s equation in the non-homogeneous case. We start with the classical Hill´s equation from which we characterize its kT -periodic solutions using classical Floquet-Lyapunov theory. After, we show how the stability-chart changes in the presence of an external periodic signal through the addition of the so-called resonance lines. This resonance lines represents new unstable regions - which are inside stable regions - in the stability-chart and to the best of the authors knowledge have not been reported in the literature. In the second part of this note we add a damping term to the Hill´s equation which causes the resonance lines disappear.