Rotating orbits of a parametrically-excited pendulum

The authors consider the dynamics of the harmonically excited parametric pendulum when it exhibits rotational orbits. Assuming no damping and small angle oscillations, this system can be simplified to the Mathieu equation in which stability is important in investigating the rotational behaviour. Analytical and numerical analysis techniques are employed to explore the dynamic responses to different parameters and initial conditions. Particularly, the parameter space, bifurcation diagram, basin of attraction and time history are used to explore the stability of the rotational orbits. A series of resonance tongues are distributed along the non-dimensionalied frequency axis in the parameter space plots. Different kinds of rotations, together with oscillations and chaos, are found to be located in regions within the resonance tongues.