Analysis of a parametrically driven pendulum

Subharmonic resonances of the parametrically driven pendulum

A simple qualitative physical explanation is suggested for the phenomenon of subharmonic resonances of a rigid planar pendulum whose axis is forced to oscillate with a high frequency in the vertical direction. An approximate quantitative theory based on the suggested approach is developed. The spectral composition of the subharmonic resonances is investigated quantitatively, and the boundaries ...

Well-behaved dynamics in a parametrically damped pendulum.

Multiple transitions to chaos in a damped parametrically forced pendulum.

We study bifurcations associated with stability of the lowest stationary point (SP) of a damped parametrically forced pendulum by varying ω0 (the natural frequency of the pendulum) and A (the amplitude of the external driving force). As A is increased, the SP will restabilize after its instability, destabilize again, and so ad infinitum for any given ω0. Its destabilizations (restabilizations) ...

Parametrically driven dark solitons.

We show that unlike the bright solitons, the parametrically driven kinks are immune from instabilities for all dampings and forcing amplitudes; they can also form stable bound states. In the undamped case, the two types of stable kinks and their complexes can travel with nonzero velocities.

Complicated Regular and Chaotic Motions of the Parametrically Excited Pendulum

Several new types of regular and chaotic behavior of the parametrically driven pendulum are discovered with the help of computer simulations. A simple physical explanation is suggested to the phenomenon of subharmonic resonances. The boundaries of these resonances in the parameter space and the spectral composition of corresponding stationary oscillations are determined theoretically and verifi...