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The Nonlinearly Damped Oscillator

Published online by Cambridge University Press:  15 September 2003

Juan Luis Vázquez
Juan Luis Vázquez*
Affiliation:
Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049 Madrid, Spain; juanluis.vazquez@uam.es.
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Abstract

We study the large-time behaviour of the nonlinear oscillator \[ \hskip-20mm m\,x'' + f(x') + k\,x=0\,, \] where m, k>0 and f is a monotone real function representing nonlinear friction. We are interested in understanding the long-time effect of a nonlinear damping term, with special attention to the model case  $f(x')= A\,|x'|^{\alpha-1}x'$  with α real, A>0. We characterize the existence and behaviour of fast orbits, i.e., orbits that stop in finite time.

Keywords

Nonlinear oscillatornonlinear dampingfast orbits.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 9 , January 2003 , pp. 231 - 246
Copyright
© EDP Sciences, SMAI, 2003

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