Given an m-parameterized family of n-dimensional vector fields, such that: (i) for some value of the parameters, the family has an equilibrium point, (ii) its linearization has a double zero eigenvalue and no other eigenvalue on the imaginary axis, sufficient conditions on the vector field are given such that the dynamics on the two-dimensional center manifold is locally topologically equivalent to the versal deformation of the planar Takens–Bogdanov bifurcation.
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.
We consider the macroscopic, second order model of Kerner-Konhäuser for traffic flow given by a system of PDE. Assuming conservation of cars, traveling waves solution of the PDE are reduced to a dynamical system in the plane. We prove that under generic conditions on the so-called fundamental diagram, the surface of critical points has a fold or cusp catastrophe and each fold point gives rise to a Takens-Bogdanov bifurcation. In particular, limit cycles arising from a Hopf bifurcation give place to traveling wave solutions of the PDE.
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