Using feedback control and Newton iterations to track dynamically unstable phenomena in experiments

If one wants to explore the properties of a dynamical system systematically one has to be able to track equilibria and periodic orbits regardless of their stability. If the dynamical system is a controllable experiment then one approach is a combination of classical feedback control and Newton iterations. Mechanical experiments on a parametrically excited pendulum have recently shown the practical feasibility of a simplified version of this algorithm: a combination of time-delayed feedback control (as proposed by Pyragas) and a Newton iteration on a low-dimensional system of equations. We show that both parts of the algorithm are uniformly stable near the saddle-node bifurcation: the experiment with time-delayed feedback control has uniformly stable periodic orbits, and the two-dimensional nonlinear system which has to be solved to make the control non-invasive has a well-conditioned Jacobian.