We investigate an inverted pendulum on a cart subject to a delayed feedback control force which tries to balance the pendulum. This is modelled by a two-dimensional system of delay-differential equations and can be considered as a prototype system for control problems arising in mechanical engineering. The linear stability analysis shows that there is only a bounded region of linear stability of the origin (corresponding to successful balancing), and identifies a singularity of codimension three as the organizing center for all dynamics of small amplitude.
Here we present the numerical bifurcation analysis of the ordinary differential equation governing the dynamics on the three-dimensional center manifold. This is compared directly with a bifurcation study of the full delay system in the vicinity of the singularity.
Additional information: Later published in EQUADIFF: Proceedings of the International Conference on Differential Equations, by World Scientific Publishing (2005), pp. 768-773, ISBN 9812561692
Sponsorship: The research of J.S. is supported by EPSRC Grant GR/R72020/01.