### The first dynamics is not chaotic

The rotation around an intermediate axis was analysed by [Ashbaugh et al., J. Dyn. Diff. Eq. 3 (1), pp. 67–85](//dx.doi.org/10.1007/BF01049489). They describe the phenomenon using the following differential equations:

$$ \begin{align}
\dot{θ} &= - M \left( \frac{1}{I_1} - \frac{1}{I_3} \right) \sin(θ) \sin(ψ) \cos(ψ)\\
\dot{ϕ} &= \frac{M}{I_3} \sin^2(ψ) + \frac{M}{I_1} \cos^2(ψ)\\
\dot{ψ} &= \left( \frac{M}{I_2} - \frac{M \sin^2 (ψ)}{I_3} - \frac{M \cos^2(ψ)}{I_1} \right)\cos(θ)
\end{align}$$

Obviously, $ϕ$ does not appear on the right-hand side of the equation. In particular, $ϕ$ is not relevant to describe the dynamics of $θ$ and $ψ$, which can be described by a two-dimensional system (which then drives $ϕ$). As two-dimensional systems cannot exhibit chaos, this also means that the system in question **cannot be chaotic**. Instead, $θ$ and $ψ$ exhibit a periodic dynamics. This can also be seen in the video: The flipping is pretty regular.

### Why the trajectory is area-filling

As $\dot{ϕ}$ is completely governed by a periodic process, one might naïvely expect that the entire system is also periodic. However, the frequency of the dynamics of $θ$ and $ψ$ does not seem to be commensurable by the frequency of time points at which $ϕ$ is a multiple of $2π$. Therefore $ϕ \bmod 2π$ exhibits a quasi-periodic behaviour, which is indeed capable of filling a surface.