Geometric aspects of the parametrically driven pendulum

The behaviour of the parametrically driven pendulum is very complex. Therefore, a global study is carried out to cover all possible situations. The study is mainly numeric, though primary bifurcations of subharmonic motions, as well as the homoclinic intersection of the hilltop saddle, are evaluated according to the Melnikov theory. Extended use is made of the cell-to-cell mapping algorithm to evaluate attracting basins of the various periodic motions. Heteroclinic intersections are always present, independently of the excitation intensity, so that the boundaries of attracting basins are always very complicated, even below the homoclinic tangency of the hilltop saddle. The oscillator exhibits various kinds of rotating and oscillating motions. All these motions lead to chaos after a period doubling cascade. It is shown that chaos usually occurs at a much greater excitation level than at that which produces homoclinic tangency of the hilltop saddle; the greater the damping, the greater the difference. The oscillatory chaotic motion is associated with the first change in the period two Birkhoff signature.