One-dimensional conservative motion

For force minus the derivative of a potential, total kinetic plus potential energy is conserved.

A trajectory approaching a nondegenerate unstable maximum of its potential energy has a logarithmically divergent period.

Simple harmonic motion obeys $\ddot{x}+{\omega }^{2}x=0$ and has a period independent of its amplitude.

A quartic oscillator adds an anharmonic term to the quadratic potential, for example $H=(p^2+q^2)/2+ϵq^4$.

Harmonic trajectories are isochronous: all nonconstant oscillations with a fixed frequency have the same period, regardless of amplitude.