Fig 8. Epi- and hypo-trochoids constructed as a sum of two linear oscillators with angular frequencies respectively ω₁ and ω₂.

A₁ ω₂ = 6ω₁ and B₁ ω₂ = −6ω₁. Plotting ∂_t ψ_g as a function of time is governed by the oscillator with the higher frequency, in this case ω₂ (A₂ and B₂). Simulations of these patterns are presented in S6, S7 and S8 Videos. The analysis of $\frac{d{\psi }_g\left(t\right)}{dt}$ using Eq 20 does not allow to extract accurately the values of ω₁ and ω₂, but rather gives and effective oscillator.