The Nonlinearly Damped Oscillator

Juan Luis Vázquez

ESAIM: Control, Optimisation and Calculus of Variations (2010)

  • 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 (2010): 231-246. <http://eudml.org/doc/90694>.

@article{Vázquez2010,
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. },
author = {Vázquez, Juan Luis},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Nonlinear oscillator; nonlinear damping; fast orbits.; nonlinear oscillator; fast ans slow orbits},
language = {eng},
month = {3},
pages = {231-246},
publisher = {EDP Sciences},
title = {The Nonlinearly Damped Oscillator},
url = {http://eudml.org/doc/90694},
volume = {9},
year = {2010},
}

TY - JOUR
AU - Vázquez, Juan Luis
TI - The Nonlinearly Damped Oscillator
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2010/3//
PB - EDP Sciences
VL - 9
SP - 231
EP - 246
AB - 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.
LA - eng
KW - Nonlinear oscillator; nonlinear damping; fast orbits.; nonlinear oscillator; fast ans slow orbits
UR - http://eudml.org/doc/90694
ER -

References

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