Bifurcation analysis of an inverted pendulum with delayed feedback control near a triple-zero eigenvalue singularity

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.