Comparison of the strange attractor butterfly form and the path traced out by the rotation vector of a mass around the intermediate rotation axis In this video,

we can see that the path traced out by the rotation vector of an object rotating around its intermediate axis traces out a path, (if the rotation vector starts out a little different from the rotation vector which doesn't change during the rotation, which is always the case, no matter how small the difference), which eventually is going to fill the sphere upon which the path is projected. Different initial vectors all create a path that will fill up the sphere.
This holds too for a the path traced out by a point representing the weather system. See this video,

The path traces out a path that accumulates in a butterfly. If we change the initial conditions of the point representing the weather, the same butterfly appears.
Are the two systems examples of the same chaos phenomenon? After all, if we change the direction of the rotation vector for the intermediate axis a little bit, the path the vector traces out on the surrounding sphere gets more and more diverted from the path traced out by the original vector. And if so, to which point in the "weather space" corresponds the rotation vector around the intermediate axis if the direction of the rotation vector is exactly the same as the direction of the intermediate axis?
 A: 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. 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. And just in case, somebody is skeptical: Simulations confirm this¹. 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 $2π$ (in a simulation¹). Therefore $ϕ \bmod 2π$ exhibits a quasi-periodic behaviour, which is indeed capable of filling a surface.

¹ Here is the source code of a Python simulation, if anybody wants to play with this system:
from jitcode import jitcode_lyap, provide_basic_symbols
from scipy.stats import sem
import numpy as np
from sympy import sin, cos, atan, sqrt

I_1, I_2, I_3 = 0.00121, 0.01638, 0.01748
E = 0.32333

θ_0 = np.random.uniform(0,0.025)
ψ_0 = np.random.uniform(0,np.pi)
φ_0 = atan( -sin(ψ_0) / cos(ψ_0) / cos(θ_0) )
M = sqrt( 2 * E / (cos(ψ_0)**2 * sin(θ_0)**2 / I_1 + cos(θ_0)**2 / I_2 + sin(ψ_0)**2 * sin(θ_0)**2 / I_3))

t, y = provide_basic_symbols()
θ,φ,ψ = y(0),y(1),y(2)

f = [
    -M * (1/I_1-1/I_3) * sin(θ) * sin(ψ) * cos(ψ),
    M / I_3 * sin(ψ)**2 + M / I_1 * cos(ψ)**2,
    ( M / I_2 - M * sin(ψ)**2 / I_3 - M * cos(ψ)**2 / I_1 ) * cos(θ)
    ]

n = len(f)
ODE = jitcode_lyap(f, n_lyap=n)
ODE.set_integrator("dopri5")
ODE.set_initial_value([θ_0, φ_0, ψ_0], 0.0)

states = []
lyaps  = []
for time in np.arange(0.01,10000,0.01):
    state, lyap, _ = ODE.integrate(time)
    states.append(state % (2*np.pi))
    lyaps.append(lyap)

np.savetxt("timeseries.dat", states)

#converting to Numpy array for easier handling
lyaps = np.vstack(lyaps)
for i in range(n):
    lyap = np.average(lyaps[10:,i])
    stderr = sem(lyaps[10:,i]) # Note that this only an estimate
    print("%i. Lyapunov exponent: % .6f ± %.6f" % (i+1,lyap,stderr))

A: When you refer to weather you are referring to the Lorenz simulation.  Real weather simulations are much more complicated, however, the differential equation used by Lorenz is similar to the differential equation describing small perturbations in rotations of a body about its intermediate axis of inertia. The two systems are not the same, but they share a common characteristic: a positive Lyapunov exponent.  This means they both have the propensity for exhibiting what is known as deterministic chaos.  There are so-called "Universality Classes" among such nonlinear systems such that phase space portraits appear identical (the butterfly in this case).  
