CONTROL-BASED CONTINUATION OF UNSTABLE PERIODIC ORBITS

We present an experimental procedure to track periodic orbits through a fold (saddle-node) bifurcation, and demonstrate it with a parametrically excited pendulum experiment where the control parameter is the amplitude of the excitation. Specifically we track the initially stable period-one rotation of the pendulum through its fold bifurcation and along the unstable branch. The fold bifurcation itself corresponds physically to the minimal amplitude that is able to support sustained rotation. Our scheme is based on a modification of time-delayed feedback in a continuation setting, and we show for an idealized model that it converges with the same efficiency as classical proportional-plus-derivative control.