We show that an inverted pendulum that is balanced on a cart by state-dependent delayed control may exhibit small chaotic motion about the upside-down position while the velocity of the cart performs a random walk. 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 is obtained by studying homoclinic bifurcations of a reduced three-dimensional vector field near a triple-zero eigenvalue singularity.
Additional information: Later published by Elsevier Science, (2004) Physica D: Non Linear Phenomena, 197(3-4), pp. 332-345. ISSN 0167-2789
Sponsorship: The research of J.S. is supported by EPSRC grant GR/R72020/01 and that of B.K. by an EPSRC Advanced Research Fellowship.
- feedback control, homoclinic tangency, balancing motion, delay differential equation, inverted pendulum, triple-zero singularity