What is Harmonic oscillator?

A system whose displacement $x$ from a stable equilibrium obeys

is a harmonic oscillator of natural angular frequency $\omega_0$ (period $T_0 = 2\pi/\omega_0$). The solution $x = A\cos(\omega_0t + \varphi)$ has an amplitude and phase fixed by the initial conditions and a period independent of the amplitude (isochronism).