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In this video,

enter image description here

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,

enter image description here

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?

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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))
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  • $\begingroup$ @Wrzlprmft-So the length of the path traced out by the rotation vector after one cycle has to be a multiple of $2\pi$ for the oscillation to be periodic (i.e. not area filling)? $\endgroup$ – descheleschilder Mar 6 '17 at 11:56
  • $\begingroup$ @descheleschilder: I am not exactly sure what you mean by this. The condition would be $$\left(\int_0^τ \dot{ϕ}(θ(t), ψ(t)) dt\right) \bmod 2π = 0,$$ where $$\dot{ϕ}(θ(t), ψ(t)) = \frac{M}{I_3} \sin^2(ψ(t)) + \frac{M}{I_1} \cos^2(ψ(t))$$ and $τ$ being the period length of the oscillation of $θ$ and $ψ$. $\endgroup$ – Wrzlprmft Mar 6 '17 at 12:05
  • $\begingroup$ @Wrzlprmft-So with the condition you gave in your last comment the rotation vector returns to the same point (wherever you put the starting point on the path traced out by the rotation vector) after one cycle, and therefore the path traced out by the rotation vector won't fill the whole sphere but remains the same every cycle? $\endgroup$ – descheleschilder Mar 6 '17 at 14:52
  • $\begingroup$ @descheleschilder: Yes. $\endgroup$ – Wrzlprmft Mar 6 '17 at 14:56
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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).

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    $\begingroup$ 1) The first system does not appear to have a positive Lyapunov exponent (see my answer). 2) The phase-space portraits of systems within the same universality class are usually not identical, but only topologically similar. $\endgroup$ – Wrzlprmft Mar 5 '17 at 14:41
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    $\begingroup$ @Wrzlprmft I accept your critique of my answer. I have only investigated the first system in a small perturbations approximation. There it appears to blow up (positive Lyapunov exponent) but as you point out in your answer the space-filling characteristics of the exact result can lead one to a false expectation. I fell into that trap. I agree with your second point also, that's why I used the words "appear identical" rather than "are identical." $\endgroup$ – Lewis Miller Mar 5 '17 at 20:04
  • $\begingroup$ @Wrzlprmft-First: Great! I can see what you mean. I assume that $\phi$ is the angle between the rotation vector and the axis about which it rotates, so the initial angle isn't important for the system to develop. Is there a trajectory that ends, after one cycle, to the same begin situation (if the distance travelled by the endpoint of the vector in one cycle is a multiple of $\pi$?)? Secondly: Isn't the path traced out by the endpoint of a two-dimensional two-legged pendulum not a case of chaos in two dimensions? Or do you need more than two variable to describe its motion? Nice edit! $\endgroup$ – descheleschilder Mar 6 '17 at 4:23
  • $\begingroup$ @descheleschilder: Your comment is misplaced and didn’t ping me for some reason. Please post follow-up comments under my answer. Regarding your question: 1) Is there a trajectory that ends, after one cycle, to the same begin situation – That would be a periodic trajectory. I would guess that it exists for some initial conditions or at least control parameters. Finding one may be very tedious though. 2) The double pendulum has four dynamical variables, e.g., the position and momentum of each leg. $\endgroup$ – Wrzlprmft Mar 6 '17 at 9:18

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