Control of the Planar Takens–Bogdanov Bifurcation with Applications

Abstract

It is well-known that on a versal deformation of the Takens–Bogdanov bifurcation is possible to find dynamical systems that undergo saddle-node, Hopf, and homoclinic bifurcations. In this document a nonlinear control system in the plane is considered, whose nominal vector field has a double-zero eigenvalue, and then the idea is to find under which conditions there exists a scalar control law such that be possible establish a priori, that the closed-loop system undergoes any of the three bifurcations: saddle-node, Hopf or homoclinic. We will say then that such system undergoes the controllable Takens–Bogdanov bifurcation. Applications of this result to the averaged forced van der Pol oscillator, a population dynamics, and adaptive control systems are discussed.