Stationary and non-stationary oscillatory dynamics of the parametric pendulum

The nonlinear dynamics of a parametrically excited pendulum is addressed. The proposed analytical approach aims at describing the pendulum dynamics beyond the simplified regimes usually considered in literature, where stationary and small amplitude oscillations are assumed. Thus, by combining complexification and Limiting Phase Trajectory (LPT) concepts, both stationary and non-stationary dynamic regimes are considered in the neighborhood of the main parametric resonance, without any restriction on the pendulum oscillation amplitudes. The advantage of the proposed approach lies in the possibility of identifying the strongly modulated regimes for arbitrary initial conditions and high-amplitude excitation, cases in which the conventionally used quasilinear approximation is not valid. The identification of the bifurcations of the stationary states as well as the large-amplitude corrections of the stability thresholds emanating from the main parametric resonance are also provided.