Regular and chaotic motions of the parametrically forced pendulum: Theory and simulations

New types of regular and chaotic behaviour of the parametrically driven pendulum are discovered with the help of computer simulations. A simple qualitative physical explanation is suggested to the phenomenon of subharmorric resonances. An approximate quantitative theory based on the suggested approach is developed. The spectral composition of the sub-harmonic resonances is investigated quantitatively, and their boundaries in the parameter space are determined. The conditions of the inverted pendulum stability are determined with a greater precision than they have been known earlier. A close relationship between the upper limit of stability of the dynamically stabilized inverted pendulum and parametric resonance of the hanging down pendulum is established. Most of the newly discovered modes are waiting a plausible physical explanation.