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On the stability of the upside–down pendulum with damping

A rigorous analysis is presented in order to show that, in the presence of friction, the upward equilibrium position of the vertically driven pendulum, with a small nonvanishing damping term, becomes asymptotically stable when the period of the forcing is below an appropriate threshold value. As a by–product we obtain an analytic expression of the solution for initial data close enough to the equilibrium position.