The nonlinearly damped oscillator

Vázquez, Juan Luis

ESAIM: Control, Optimisation and Calculus of Variations, Tome 9 (2003), pp. 231-246.

We study the large-time behaviour of the nonlinear oscillator

$$m{x}^{\text{'}\text{'}}+f\left(x^{\text{'}}\right)+kx=0,$$

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 $f\left(x^{\text{'}}\right)=A{\left|x^{\text{'}}\right|}^{\alpha -1}{x}^{\text{'}}$ with $\alpha$ real, $A>0$. We characterize the existence and behaviour of fast orbits, i.e., orbits that stop in finite time.

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