We investigate a delay differential equation that models a pendulum stabilized in the upright position by a delayed linear horizontal control force. Linear stability analysis reveals that the region of stability of the origin (the upright position of the pendulum) is bounded for positive delay. We find that a codimension-three triple-zero eigenvalue bifurcation acts as an organizing centre of the dynamics. It is studied by computing and then analysing a reduced three-dimensional vector field on the centre manifold. The validity of this analysis is checked in the full delay model with the continuation software DDE-BIFTOOL. Among other things, we find stable small-amplitude solutions outside the region of linear stability of the pendulum, which can be interpreted as a relaxed form of successful control
Original language | English |
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DOIs | |
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Publication status | Unpublished - 2003 |
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Additional information: Later published by Institute of Physics, Nonlinearity, 17(1), pp. 85-103, ISSN 0951-7715
Sponsorship: The research of J.S. is supported by EPSRC grant GR/R72020/01
Terms of use: Nonlinearity © copyright 2004 IOP Publishing Ltd.