Astroid motion of a harmonic oscillator in a rotating frame ($(x^{\prime },y^{\prime })=({\cos }^3\omega t,{\sin }^3\omega t)$)

For an isotropic harmonic oscillator with inertial frequency $2\omega$, viewed in a frame rotating at $\omega$, initial rotating coordinates $(1,0)$ and zero relative velocity produce the displayed astroid trajectory. The complex coordinate satisfies $\ddot{\zeta }+2i\omega \dot{\zeta }+3{\omega }^2\zeta =0$, with solution $\zeta =\frac{3}{4}{e}^{i\omega t}+\frac{1}{4}{e}^{-3i\omega t}$. The relative speed is $(3|\omega |/2)|\sin 2\omega t|$, attaining its first maximum at $\pi /(4|\omega |)$ for nonzero $\omega$.