The nonlinearly damped oscillator

Juan Luis Vázquez

ESAIM: Control, Optimisation and Calculus of Variations (2003)

  • Volume: 9, page 231-246
  • ISSN: 1292-8119

Abstract

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We study the large-time behaviour of the nonlinear oscillator 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 ' | α - 1 x ' with α real, A > 0 . We characterize the existence and behaviour of fast orbits, i.e., orbits that stop in finite time.

How to cite

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Vázquez, Juan Luis. "The nonlinearly damped oscillator." ESAIM: Control, Optimisation and Calculus of Variations 9 (2003): 231-246. <http://eudml.org/doc/245190>.

@article{Vázquez2003,
abstract = {We study the large-time behaviour of the nonlinear oscillator\[ \hspace\{-56.9055pt\}m\,x^\{\prime \prime \} + f(x^\{\prime \}) + k\,x=0\,, \]where $m, k&gt;0$ 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 $f(x^\{\prime \})= A\,|x^\{\prime \}|^\{\alpha -1\}x^\{\prime \}$ with $\alpha $ real, $A&gt;0$. We characterize the existence and behaviour of fast orbits, i.e., orbits that stop in finite time.},
author = {Vázquez, Juan Luis},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {nonlinear oscillator; nonlinear damping; fast orbits; fast ans slow orbits},
language = {eng},
pages = {231-246},
publisher = {EDP-Sciences},
title = {The nonlinearly damped oscillator},
url = {http://eudml.org/doc/245190},
volume = {9},
year = {2003},
}

TY - JOUR
AU - Vázquez, Juan Luis
TI - The nonlinearly damped oscillator
JO - ESAIM: Control, Optimisation and Calculus of Variations
PY - 2003
PB - EDP-Sciences
VL - 9
SP - 231
EP - 246
AB - We study the large-time behaviour of the nonlinear oscillator\[ \hspace{-56.9055pt}m\,x^{\prime \prime } + f(x^{\prime }) + k\,x=0\,, \]where $m, k&gt;0$ 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 $f(x^{\prime })= A\,|x^{\prime }|^{\alpha -1}x^{\prime }$ with $\alpha $ real, $A&gt;0$. We characterize the existence and behaviour of fast orbits, i.e., orbits that stop in finite time.
LA - eng
KW - nonlinear oscillator; nonlinear damping; fast orbits; fast ans slow orbits
UR - http://eudml.org/doc/245190
ER -

References

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