# The Nonlinearly Damped Oscillator

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

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

## Access Full Article

top## Abstract

top## How to cite

topVázquez, 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

top- S. Angenent and D.G. Aronson, The focusing problem for the radially symmetric porous medium equation. Comm. Partial Differential Equations20 (1995) 1217-1240.
- D.G. Aronson, The Porous Medium Equation. Springer-Verlag, Berlin/New York, Lecture Notes in Math.1224 (1985).
- D.G. Aronson, O. Gil and J.L. Vázquez, Limit behaviour of focusing solutions to nonlinear diffusions. Comm. Partial Differential Equations23 (1998) 307-332.
- D.G. Aronson and J. Graveleau, A selfsimilar solution to the focusing problem for the porous medium equation. Euro. J. Appl. Math.4 (1992) 65-81.
- D.G. Aronson and J.L. Vázquez, The porous medium equation as a finite-speed approximation to a Hamilton-Jacobi equation. Ann. Inst. H. Poincaré Anal. Non Linéaire4 (1987) 203-330.
- H. Brezis, L.A. Peletier and D. Terman, A very singular solution of the heat equation with absorption. Arch. Rational Mech. Anal.95 (1986) 185-209.
- J. Carr, Applications of centre manifold theory. Springer-Verlag, New York-Berlin, Appl. Math. Sci. 35 (1981) vi+142 pp.
- M. Chaves and V. Galaktionov, On the focusing problem for the PME with absorption. A geometrical approach (in preparation).
- J.I. Díaz, Nonlinear partial differential equations and free boundaries. Vol. I. Elliptic equations. Pitman (Advanced Publishing Program), Boston, MA, Res. Notes in Math.106 (1985).
- J.I. Díaz and A. Li nán, On the asymptotic behaviour for a damped oscillator under a sublinear friction. Rev. Acad. Cien. Ser. A Mat.95 (2001) 155-160.
- R. Ferreira and J.L. Vázquez, Self-similar solutions to a very fast diffusion equation. Adv. Differential Equations (to appear).
- V.A. Galaktionov, S.I. Shmarev and J.L. Vázquez, Second-order interface equations for nonlinear diffusion with very strong absorption. Commun. Contemp. Math.1 (1999) 51-64.
- V.A. Galaktionov, S.I. Shmarev and J.L. Vázquez, Behaviour of interfaces in a diffusion-absorption equation with critical exponents. Interfaces Free Bound.2 (2000) 425-448.
- V.A. Galaktionov, S.I. Shmarev and J.L. Vázquez, Regularity of interfaces in diffusion processes under the influence of strong absorption. Arch. Ration. Mech. Anal.149 (1999) 183-212.
- J. Guckenheimer and Ph. Holmes, Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. Revised and corrected reprint of the 1983 original. Springer-Verlag, New York, Appl. Math. Sci. 42 (1990).
- A. Haraux, Comportement à l'infini pour certains systèmes non linéaires. Proc. Roy. Soc. Edinburgh Ser. A84 (1979) 213-234.
- M.W. Hirsch and S. Smale, Differential equations, dynamical systems, and linear algebra. Academic Press, New York-London, Pure Appl. Math. 60 (1974).
- S. Kamin, L.A. Peletier and J.L. Vázquez, A nonlinear diffusion-absorption equation with unbounded initial data, in Nonlinear diffusion equations and their equilibrium states, Vol. 3. Gregynog (1989) 243-263. Birkhäuser Boston, Boston, MA, Progr. Nonlinear Differential Equations Appl.7 (1992).
- E.B. Lee and L. Markus, Foundations of Optimal Control Theory. J. Wiley and Sons, New York, SIAM Ser. Appl. Math. (1967).
- O.A. Oleinik, A.S. Kalashnikov and Y.-I. Chzou, The Cauchy problem and boundary problems for equations of the type of unsteady filtration. Izv. Akad. Nauk SSR Ser. Mat.22 (1958) 667-704.
- L. Perko, Differential equations and dynamical systems, Third edition. Springer-Verlag, New York, Texts in Appl. Math. 7 (2001).
- J.L. Vázquez, An Introduction to the Mathematical Theory of the Porous Medium Equation, in Shape Optimization and Free Boundaries, edited by M.C. Delfour. Kluwer Ac. Publ., Dordrecht, Boston and Leiden, Math. Phys. Sci. Ser. C 380 (1992) 347-389.

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.