Control of the Planar Takens-Bogdanov Bifurcation with Applications

• Autores: Francisco A. Carrillo, Fernando Verduzco
• Localización: Acta applicandae mathematicae, ISSN 0167-8019, Vol. 105, Nº. 2, 2009, págs. 199-225
• Idioma: inglés
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• Resumen

  • 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.