Controllability near Takens-Bogdanov points
The authors study controllability properties of the control-affine system whose uncontrolled part coincides with a universal unfolding of the Takens-Bogdanov singularity \[ {dx\over dt}=y,\;{dy\over dt}= \lambda_1 y+\lambda_2 x+x^2+ xy+u(t), \tag{*} \] where \(u\in U^\rho: =\{u\in L^\infty( \mathbb{R},\mathbb{R}):\text{ess sup}|u(t) |\leq\rho\}\), \(\rho\geq 0\). The main result is that the qualitative behavior of control system (*) can be different from the behavior of all systems with constant control functions in the following sense. There are parameter regions and control ranges such that for constant controls there is no homoclinic orbit, while there exists a ``controlled homoclinic'' orbit corresponding to a nonconstant control function.
scientific article; zbMATH DE number 1431824