Anharmonic oscillator

Remarks

The pendulum equation $\theta'' + \frac{g}{\ell} \sin \theta = 0$ is anharmonic: here $m = 1$ and $V(\theta) = \frac{g}{\ell} (1 - \cos \theta)$ on $I = (-\pi, \pi)$, which is not quadratic since $V(\theta) = \frac{g}{2\ell} \theta^2 - \frac{g}{24 \ell} \theta^4 + O(\theta^6)$. The undamped, unforced Duffing equation $x'' + \alpha x + \beta x^3 = 0$ with $\alpha > 0$ and $\beta \neq 0$ is anharmonic too, with $V(x) = \frac{\alpha}{2} x^2 + \frac{\beta}{4} x^4$. The harmonic oscillator $m x'' + k x = 0$ is not anharmonic: its potential is exactly $\frac{k}{2} x^2$.

The terms of order three and higher in the Taylor expansion of $V$ at $0$ are called the anharmonic terms. Solutions of small amplitude oscillate with angular frequency close to $\sqrt{k/m}$, and in general the period depends on the amplitude.