Intermediate energy

At higher energy, more sets of initial conditions produce chaos. The animation and Poincare section below are for E = 8. A large part of the Poincare section is filled with scattered points. However, some regular behaviour (corresponding to curves and points in the section) is also observed, and the animation provides an example of regular behaviour. The animation indicates that the pendulum is not particularly energetic: in fact there is insufficient energy for the lower plate to rotate completely (this requires energies larger than E = 17.0, approximately).

The motion is almost completely chaotic around E = 20 (the Poincare section is almost completely filled with points, although it is notable that the slow mode survives). The animation and Poincare section below illustrate this case. The lower plate now has sufficient energy to rotate completely, but the upper plate cannot rotate completely (this happens for energies larger than E = 37.9, approximately). .

The motion is completely chaotic around E = 35 (the Poincare section is filled with dots, within the allowed region). The upper plate still has insufficient energy to rotate completely. The animation and Poincare section below illustrate this case.

Both plates can rotate completely when the energy is larger than about E = 54.9. At somewhat higher energy regular behaviour begins to reappear in the Poincare section. The animation and Poincare section below illustrate the case E = 80.

The dynamics of the pendulum at low energy is described here. The dynamics of the pendulum at high energy is described here. A large gallery of Poincare sections is provided here.

Main page on the double square pendulum
Modelling and numerical solution
Dynamics 1: Poincare sections and animations for low energy
Dynamics 3: Poincare sections and animations for high energy
Images and videos of the device
Simple double pendulum page
Mike Wheatland's home page