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RECENTLY, Anne Skeldon and Tom Mullin of the Nonlinear Systems Group
at the Clarendon laboratory, Oxford, collaborated with David Pottinger at
the IBM UK Scientific Centre in Winchester to investigate the best way of
using three-dimensional computer graphics to make phase portraits.

The project involved studying the motions of a simple system of coupled
pendula. One pendulum hangs down, and is allowed to swing from side to side.
The second pendulum is attached to the bottom of the first pendulum but
swings in a direction at right angles to the first. The pivot point of the
top pendulum is driven up and down in a regular way by a motor at a certain
frequency (see left). We were interested in studying the patterns produced
as the two pendula moved around, when the frequency of the motor was changed.

Like the astronautical system that Ian Stewart mention in the main text,
our system is a complex assembly of periodic motions. These are described
in terms of five coordinates. They are the velocity and angle of the top
pendulum, the velocity of the drive and the velocity and angle of the bottom
pendulum.

To obtain the phase portraits, we took the differential equations describing
the periodic motions of the pendula, then put in starting values for all
the coordinates and calculated the values for a later time on a computer.
The computer repeats the process through a series of time-steps until it
has an adequate numerical description of the system’s dynamic behaviour.
In other words, we have access to the complete trajectories of all five
coordinates.

The computer now has enough information to provide a phase portrait.
Because there are five coordinates, the phase portrait must be in five-dimensional
phase space. The next trick was to find the best way of displaying five
dimensions on a two-dimensional computer screen. We chose a method that
highlighted the important qualitative features.

Graphics a to b show the projections obtained by picking three of the
five coordinates and plotting the resulting trajectories. What we get are
two attractors for the system. Each attractor is a torus, coloured differently
to indicate that they are distinct. Both attractors are for the same driving
frequency of the motor. The one that we obtain depends on the starting values
of the coordinates.

The apparent intersection of the two tori is an artefact of the projection,
because, remember, the system is in five dimensions. If we change the driving
frequency slightly, the tori become distorted and move closer together as
shown in Graphic b. On reducing the frequency further, the two separate
tori join through a ‘gluing’ bifurcation to become one (monochrome) attractor
as in Graphic c.

To understand the five-dimensional attractor better, we assigned colours
to the trajectories to indicate a fourth dimension. In Graphic d, we chose
the projection of the attractor in the same way as above and coloured it
according to the sign of the fourth coordinate. This tells us on which part
of the cycle the pendulum changes direction.

Graphic e is a similar attractor, but is coloured according to the magnitude
of the fourth coordinate. The green band indicates where one of the pendula
is moving more slowly and successfully shows that the ‘gluing’ process happens
in this part of the motion. The Graphic on the front-cover shows the two
separate tori, coloured in the same way as in Graphic e, just before the
‘gluing’ process has taken place.

Finally, we show in Graphic f, a Poincare section of the joined torus;
notice that the surface is smooth. If, however, we change the driving frequency
by a tiny amount, the motion becomes chaotic, as shown in Graphic g and
the torus begins to break up. Thus there is no longer a smooth surface associated
with the phase portrait and representing it will require some interesting
new graphical techniques. Anne Skeldon

Ian Stewart is a mathematician in the University of Warwick’s Nonlinear
Systems Laboratory. His recent popular book on chaos, Does God Play Dice?
is published by Basil Blackwell.

Further reading: ‘Chaos’, Scientific American, December 1986, p38; Chaotic
Dynamical Systems will describe Robert L. Devaney, Addison-Welsey, Redwood
City California, 1989.

Next week: Tom Mullin will show how phase portraits can model turbulence
in fluids, and Tim Palmer will describe how the Lorenz attractor and powerful
computer models improve our understanding of weather and climate.