Rotational Number Approach to a Damped Pendulum under Parametric Forcing

We study the dynamical behavior of a damped pendulum under parametric forcing, which exhibits various chaotic dynamics characterized by the rotation number: oscillating, rotating, and tumbling chaos. The analysis of the detailed bifurcation diagram together with the rotation number reveals the existence of multiple types of rotational onset. At relatively high forcing frequencies, the system successively undergoes a hysteretic rotational onset from oscillating chaos to periodic rotation due to bistability and a non-hysteretic onset from rotating chaos to tumbling chaos. The onset mechanism of the latter is found to result from an interior crisis and an attractor merging crisis. On the other hand, at relatively low forcing frequencies the system exhibits a direct non-hysteretic onset from periodic oscillation to tumbling chaos, arising from a tangency crisis. This reveals the complex structure of the phase diagram at low forcing frequencies.