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

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

2009-03-18

arxiv.org/abs/0903.3144v1

Mathematics

Dynamical Systems

6 pages, 3 figures

Scientific paper

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.

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