Bifurcation Theory In Mechanical Systems.

Abstract

In this paper, bifurcation theory is used to classify di􀀞erent dynamical behaviors occurring
in a mechanical system under bounded control actions. The example is a pendulum
with an inertia disc mounted in its free extreme. By design, the control action can only be
introduced by means of an external torque applied by a DC motor to the inertia disc. Imposing
a bounded control action places an important obstacle to the design of a controller
capable to drive the pendulum from rest to the inverted position and to stabilize it there.
The only way in which the pendulum can reach the inverted position is by oscillations
of increasing amplitudes. Due to the saturation of the control law the trivial equilibrium
points -the rest and the inverted position- experiment a pitchfork bifurcation when one
key parameter is varied. Therefore, two additional equilibrium points associated to each
equilibrium of the non-forced system do appear. If another control parameter is varied,
homoclinic and heteroclinic bifurcations, saddle-node bifurcations of periodic orbits, and
Hopf bifurcations of equilibria do appear. Some of these codimension one bifurcations
are organized in a codimension two Bogdanov-Takens bifurcation, when varying two parameters
simultaneously. The application of both numerical and analytical tools from
bifurcation theory to understand and classify the dynamical behavior of the closed-loop
system facilitates the control law design, as shown in the paper.