Complex balancing motions of an inverted pendulum subject to delayed feedback control

We show that an inverted pendulum that is balanced on a cart by linear delayed feedback control may exhibit small chaotic motion about the upside-down position. In periodic windows associated with this chaotic regime we find periodic orbits of arbitrarily high period that correspond to complex balancing motion of the pendulum with bounded velocity of the cart. This result shows that complex balancing is possible in a controlled mechanical system with a geometric nonlinearity even when the control law is only linear. This is in contrast to other proposed models that require a nonlinearity of the controller, such as round-off due to digitization. We find the complex motion by studying homoclinic bifurcations of a reduced three-dimensional vector field near a triple-zero eigenvalue singularity. More generally, the dynamics we describe must be expected in any system with a triple-zero singularity and reflection symmetry.