On the weakly damped harmonic oscillator

The initial-value problem for the differential equation \[ u”\left ( t \right ) + \epsilon \left [ {a + {{\left ( {u’} \right )}^2}} \right ]u’ + u = 0\] is studied under the assumption that $a$ and $\epsilon$ are positive constants and $\epsilon$ is small. Perturbation methods are used to obtain a first approximation to the solution that is uniformly valid in $t$. For any non-zero $a$ the solution ultimately decays exponentially with a time scale ${\left ( { \epsilon a} \right )^{ - 1}}$. For sufficiently small $a$, however, the damping is dominated by the cubic damping term for times of order ${ \epsilon ^{ - 1}}$ and is thus algebraic in character. The frequency of the oscillations is reduced by an amount $\frac {1}{8}{\left ( { \epsilon a} \right )^2}$ and is unaffected by the cubic damping to the order of approximation considered.