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An enhanced and more exact criterion for dynamic stabilization of the parametrically driven inverted pendulum is obtained: the boundaries of stability are determined with greater precision and are valid in a wider region of the system parameters than previous results. The lower boundary of stability is associated with the phenomenon of subharmonic resonances in this system. The relationship of the upper limit of dynamic stabilization of the inverted pendulum with ordinary parametric resonance (i.e. with destabilization of the lower equilibrium position) is established. Computer simulation of the physical system aids the analytical investigation and proves the theoretical results.
Eugene I. Butikov (Fri,) studied this question.