Nonlinear dynamics of a rotating double pendulum

• The rotating double pendulum shows transitions from chaos to order and back to chaos.
• These transitions occur as the rotation speed is increased.
• The dynamics is quasi-periodic in the ordered state.
• Within the ordered state the nature of quasi-periodicity changes with rotation speed.
• The chaotic state always emerges as a result of an internal resonance.

Nonlinear dynamics of a double pendulum rotating at a constant speed about a vertical axis passing through the top hinge is investigated. Transitions of oscillations from chaotic to quasiperiodic and back to chaotic again are observed with increasing speed of rotation. With increasing speed, a pair of new stable equilibrium states, different from the normal vertical one, appear and the quasiperiodic oscillations occur. These oscillations are first centered around the origin, but with increasing rotation speed they cover the origin and the new fixed points. At a still higher speed, more than one pair of fixed points appear and the oscillation again turns chaotic. The onset of chaos is explained in terms of internal resonance. Analytical and numerical results confirm the critical values of the speed parameter at various transitions.