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

TL;DR

This paper presents a method combining feedback control and Newton iterations to track unstable phenomena in experiments, demonstrated on a parametrically excited pendulum, ensuring stability near bifurcations.

Contribution

It introduces a practical approach integrating time-delayed feedback control with Newton iterations for experimental bifurcation analysis, showing stability near saddle-node bifurcations.

Findings

Stable periodic orbits with time-delayed feedback control.

Well-conditioned Jacobian for Newton iteration.

Feasibility demonstrated on a parametrically excited pendulum.

Abstract

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.