The Transition to Chaos of Pendulum Systems

We examine the nonlinear response of two planar pendula under external and kinematic excitations, which are very relevant as paradigmatic models in nonlinear dynamics. These pendula act under the action of an additional constant torque, and are subjected to one of the following excitations: a further external periodic torque, and a vertically periodic forcing of the point of suspension. Here, we show the influence of the constant torque strength on the transition to chaotic motions of the pendulum using both Melnikov analysis and the computation of the basins of attraction. The global bifurcations are illustrated by the erosion of the corresponding basins of attraction.