Controllability near Takens-Bogdanov points
type: Item
instance of: scholarly article
title: Controllability near Takens-Bogdanov points
zbMATH Open document ID: 0953.93040
DOI: 10.1007/BF02254704
author: G. Häckl
author: Klaus R. Schneider
published in: Journal of Dynamical and Control Systems
publication date: 2000-04-13T00:00:00Z
review text: 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.
reviewed by: He Chungyou
Mathematics Subject Classification ID: 93C15
Mathematics Subject Classification ID: 34C37
Mathematics Subject Classification ID: 34C23
Mathematics Subject Classification ID: 34H05
zbMATH DE Number: 1431824
zbMATH Keywords: control set
zbMATH Keywords: bifurcation
zbMATH Keywords: homoclinic orbit
zbMATH Keywords: Takens-Bogdanov singularity
zbMATH Keywords: qualitative behavior
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1431824