In the wonderful Deutsches Museum of Science and Technology in Munich, Germany, there is a splendid display of a real physical double pendulum that exhibits dramatically chaotic motion. You can watch a video clip I took on my visit. The pendulum is contained in a clear plastic frame, with the central shaft emerging. A visitor is able to start the chaotic motion by giving the shaft a good twist and imparting some angular momentum.

The geometry of a somewhat idealized double pendulum that is more readily analyzed is:

Rotating around the central free pivot there are now two distinct
masses, m_{1} and m_{2}, with the inner mass
m_{1} located on a second freely rotating pivot. Instead of just one length, now
the lengths of two segments of massless arm,
l_{1} and l_{2}, are adjustable parameters.
Since the dynamic state of this physical system is now dependent on two variable
angles, there will be two simultaneous differential equations required to completely
characterize the motion. The differential equations of motion may best be
derived by the Lagrangian method. The Lagrangian *L* is defined as
*L* = *T* - *V*, where *T* is the sum of the kinetic energy of the two
masses, and *V* is their total potential energy. For the system pictured above, this can be
derived as

From this definition the Euler-Lagrange equation is applied twice with respect
to the two angle variables θ_{1} and θ_{2}
to derive the two simultaneous equations of motion:

After some additional derivation, these may be put in a form that can be
solved numerically to simulate the time dependence of the two angles. There are four
initial conditions that must be specified, the two initial angles
θ_{1} and θ_{2} and the two angular velocities
*d*θ_{1}/*dt* and *d*θ_{2}/*dt*.
Note that in the physical museum pendulum above, only *d*θ_{1}/*dt*
can be given a nonzero initial transient, and the angle θ_{2}
is unreachable.

Due to the extra flexibility of the additional pivot, stable simple harmonic motion
is not possible for this system, but both quasi-periodic and chaotic motion are
readily observable. As in the physical museum pendulum, initially giving a large
*d*θ_{1}/*dt* and a small initial *d*θ_{2}/*dt*
will tend to lead to chaotic behavior. Assuming it is possible, as in the
theoretical system in the figure above, giving both
*d*θ_{1}/*dt* and *d*θ_{2}/*dt*
similar initial values will tend to initiate stable quasi-static behavior.

As examples, the following are some simulated plots for two different sets of initial
conditions. Since there are four state variables in this dynamical system, and four
dimensional plotting is not possible to visualize, we cannot present a full state
diagram. However, just looking at two of the state variables, namely the two angular
velocities, a marked difference between quasi-periodic and chaotic modes is readily
apparent. Consider first setting the initial conditions of both
*d*θ_{1}/*dt* and *d*θ_{2}/*dt*
to 10 radians/second. The following plot shows the regular pattern of
quasi-static motion:

These initial conditions cause the motion of the double pendulum to be dominated by both masses rotating around and around the center pivot at a common angular velocity, with a little flexing or wobbling of the second pivot.

As in the case of the single pendulum, we can generate an audio track that corresponds to this motion by using the simulated angular velocities of the two masses as two audio signals, sampled at a much higher rate in time. In the case of the single pendulum this gave a monaural audio track, but since there are two angular velocities for the double pendulum, these can be arranged into a stereo track:

Note that with the quasi-period motion the corresponding sound track is dominated
by discrete tones. Another way to induce quasi-periodic motion is to only spin the
inner angle initially, and at a low velocity, so the pendulum will just wobble back
and forth. This stereo audio track corresponds to the initial conditions of
*d*θ_{1}/*dt*=5 radians/second and
*d*θ_{2}/*dt*=0:

In contrast, setting initial values of *d*θ_{1}/*dt*=10 radians/second
but *d*θ_{2}/*dt*=0 demonstrates a chaotic plot of
completely different character:

As before we can digitally compose a stereo audio track by sampling the two angular velocities, and for this chaotic case results in a sound pattern of a completely atonal character:

It is also interesting to see two different state variables plotted against
each other, the two displacement angles θ_{1} and θ_{2} in
chaotic mode:

Run from a shell or command terminal in the download directory with:

java -jar DoublePendulum.jar

Note that two parameters you are able to adjust in this application are called the
"mass fraction" and "length fraction". The mass fraction α,
a value between 0 and 1, is defined to be
m_{2} / (m_{1} + m_{2}), or in other words, the fraction of the
total mass that is associated with the outer weight. Similarly the length fraction β,
also a value between 0 and 1, is defined to be
l_{2} / (l_{1} + l_{2}), or the fraction of the total pendulum
length between the inner and outer masses. The values of these two parameters are
a large determining factor whether the pendulum motion is quasi-periodic or chaotic.
A small or large length fraction value, placing the inner weight
toward either inner or outer end, tends to stabilize the motion
into quasi-periodic mode, whereas length fractions around 0.5 tend to lead to chaotic
motion.

You may adjust the various parameters controlling the simulations by either moving the sliders or by typing in a numerical value and hitting the Enter key.

To study the range of values of the two parameters α and β that lead to
quasi-periodic or chaotic behavior, it is convenient to have a numerical measure that
indicates the class of motion that may be plotted with respect to the parameter values.
One easily computed metric is the
Weiner entropy, or
spectral flatness metric. If we take the Fourier transform of one of the mechanical
measures of the double pendulum, for example, the angular velocity of the
first member *d*θ_{1}/*dt*, the spectrum will consist of
only a series of spikes for quasi-periodic motion, but will spread out into a noisy base
under chaotic conditions. The entropy of the spectrum will be low, close to 0, when its energy is
concentrated in a few spikes, but much higher, close to 1, for noisier spectra.

As examples of how this measure predicts the regions of chaotic and quasi-periodic pendulum behavior, consider the first plot:

This shows the spectral entropy of the angular velocity of θ_{1}
plotted versus β, with α set to the equal masses value of 0.5. Note that chaotic
motion, associated with high entropy values, is only present for length ratios in the vicinity 0.5,
the case of equal arm lengths. In contrast, the second entropy plot:

plotting entropy versus α with β set to 0.5 for equal lengths, shows that the presence of chaotic motion over a much wider range of the mass ratio.

The final plot is a three dimensional projection of the entropy as both α and β are varied over their practical ranges of 0.1 to 0.9:

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