On periodic differential equations with dissipation

In this work we present an unexpected relation between the discriminant associated to a Hill equation with and without dissipation. We prove that by knowing the discriminant associated to a periodic differential equation, which is the summation of the monodromy matrix main diagonal entries, we are able to obtain the stability properties of damped periodic differential equation solutions. We propose to conceive the discriminant as a manifold, by doing this one can observe that the stability properties of periodic differential equations are closely related to the growing rate of unstable solutions of periodic differential equations without dissipation. We show the appearance of the Ziegler destabilization paradox in systems of one degree of freedom. This work may be of interest for scientists and engineers dealing with parametric resonance applications or physicist working on the motion of a damped wave in a periodic media.