Summary: The second-order linear differential equation
is investigated, where , and . We say that a parametric resonance occurs in this equation, if for every sufficiently small there are such that the equation has solutions with amplitudes tending to , as . The period of the parametric excitation is called a critical value of the parametric resonance if with some for all sufficiently small . We give a new simple geometric proof for the fact that the critical values are the natural numbers. We apply our method also to find the most effective control destabilizing the equilibrium , and to give a sufficient condition for the parametric resonance in the asymmetric case .