Codimension-two bifurcations in a pendulum with feedback control

We study non-linear dynamics and bifurcation behavior of a pendulum with position and velocity feedback control. The pendulum is assumed to be driven by a servo-motor to which electromotive force proportional to both the difference of the pendulum position from the desired one and to the pendulum velocity is applied. The servo-motor dynamics is modeled by a first-order differential equation, so that the full system is of third order. Using the center manifold and normal form theories, we analyze three types of codimension-two bifurcations. Our analysis reveals that such bifurcations can occur at infinitely many points in the two-parameter space. Numerical examples are given with some simulation results to illustrate the theoretical results. Seemingly infinite sequences of saddle-node bifurcations for periodic orbits are also observed in the numerical simulations.