Geoff,
     The pendulum responsible for the now-famous picture you and Meredith mention (and give the American Physical Society-hosted sebsite) is not like the dangling pendulum we've been discussing.
    As you noted, it is a 'toy'-version of the one that I studied (responsible for the aforementioned article on chaos) and which has for years been used to generate beautiful art-pieces.
A picture of that pendulum which I built (while still at Texas Tech University) and which is here in the Mercer physics department is to be found at
http://physics.mercer.edu/Science_Art/bowling_ball.htm
       Also on this website are some of the myriad traces that have been produced with the pendulum.  Art-folks love these because of their 'life-like' properties--no two exactly the same (quite unlike computer generated patterns).
   The bowling ball is supported by a pipe on one end, the other end of which is connected to a universal joint (off a toyota pickup if I remember correctly).  Because of this U-joint, the x-y axes
are coupled, giving rise to a very large number of different patterns during free decay.  The moments of inertia in the
two axes are adjustable to provide additional control that assists the near infinity of possible results.  In the absence of  friction, this instrument would display what's called Hamiltonian chaos.  The sensitive dependence on initial
conditions (essence of the butterfly effect) means that long term prediction (just like the weather) is
unpredictable.  There are two capacitive sensors that map the motion of the bowling ball, one in each of the perpendicular axes.  The output from the amplified signals is fed to an x-y (analog recorder) that with a ball-point
pen generates the images on ordinary paper.
   I have done a lot of chaos research in the last 15 years.  The chaotic pendulum that I designed and which is
online here at Mercer (when my colleague Matt Marone gets it back operational after our move to the new
building) is at
   http://physics.mercer.edu/PENDULUM/
This pendulum can be controlled from anywhere in the world over the internet when operational.  The parameter
that the user can adjust is the frequency of the drive.  The pendulum's motion is monitored by one of my SDC
sensors functioning as a velocity detector.  An aluminum disk rotates with the pendulum between rare-earth
magnets positioned on either side, which are on the end of a bending cantilever.  The output (velocity) is integrated
with a 'leaky' integrator to give the position.
    Largely because of my pendulum chaos studies I was asked and wrote an article on " math methods used by physicists to study chaos" for the 10th Ed. of the McGraw Hill Encyclopedia of Science and Technology.  I'm sure that some readers will note the prominence of other authors in the chaos section of the encylopedia (such as the creator of the butterfly effect, Ed Lorenz) and then will say, "but who's this guy, Peters?".
     You ask, Geoff, 'why use multiple turn coils for the Faraday-law sensor.  The answer is resident
within the statement of his law (generated, in the minds of most, by the greatest experimentalist who ever
lived).  Faraday's law states that the voltage generated within the wire (historically called the electromotive force (emf) even though voltage is not a force) is proportional to the number of turns of the coil times the time rate of change of the magnetic flux passing through the coil.  Thus no matter how your amplifier is built (solid state or vacuum tubes) the signal will be greater the larger number of turns you can wind within the constraints of space limitations borne of wire size and increased resistance that
results when the wire gets too small trying to put more turns into a given place.
   Randall