Control-Based Continuation of Unstable Periodic Orbits

Author:

Sieber Jan1,Krauskopf Bernd2,Wagg David3,Neild Simon3,Gonzalez-Buelga Alicia4

Affiliation:

1. Department of Mathematics, University of Portsmouth, PO1 3HF Portsmouth, UK

2. Department of Engineering Mathematics, University of Bristol, Bristol BS8 1TR, UK

3. Department of Mechanical Engineering, University of Bristol, Bristol BS8 1TR, UK

4. Department of Civil Engineering, Universitat Politècnica de Catalunya, 08034 Barcelona, Spain

Abstract

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 tracking 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 to the minimal amplitude that supports 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.

Publisher

ASME International

Subject

Applied Mathematics,Mechanical Engineering,Control and Systems Engineering,Applied Mathematics,Mechanical Engineering,Control and Systems Engineering

Reference20 articles.

1. Lecture Notes on Numerical Analysis of Nonlinear Equations;Doedel

2. Elements of Applied Bifurcation Theory;Kuznetsov

3. MatCont: A Matlab Package for Numerical Bifurcation Analysis of ODEs;Dhooge;ACM Trans. Math. Softw.

4. Engelborghs, K., Luzyanina, T., and Samaey, G., 2001, “DDE-BIFTOOL v.2.00: A Matlab Package for Bifurcation Analysis of Delay Differential Equations,” Report No. TW 330, Katholieke Universiteit Leuven.

5. An Adaptive Newton-Picard Algorithm With Subspace Iteration for Computing Periodic Solutions;Lust;SIAM J. Sci. Comput. (USA)

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