Local bifurcation analysis in the Furuta pendulum via normal forms

Inverted pendulums are very suitable to illustrate many ideas in automatic control of nonlinear systems. The rotational inverted pendulum is a novel d esign that has some interesting dynamics features that are not present in i nverted pendulums with linear motion of the pivot. In this paper the dynami cs of a rotational inverted pendulum has been studied applying well-known r esults of bifurcation theory. Two classes of local bifurcations are analyze d by means of the center manifold theorem and the normal form theory - firs t, a pitchfork bifurcation that appears for the open-loop controlled system ; second, a Hopf bifurcation, and its possible degeneracies, of the equilib rium point at the upright pendulum position, that is present for the contro lled closed-loop system. Some numerical results are also presented in order to verify the validity of our analysis.