Control of oscillations from the k-zero bifurcation

In this paper, an analytical method for the analysis and control of oscillations in non-linear control systems, whose linearization around the origin has k eigenvalues zero, is presented. The main idea consists in exploit, for the particular case of the double-zero (or Takens–Bognadov) bifurcation, the existence of a curve of Hopf bifurcation points on its versal deformation, to control oscillations. Then the general case is reduced to the double-zero case through a change of coordinates and a change in the input control. The method is illustrated with the pendubot, an underactuated robot manipulator of two degrees of freedom.