38 ATaleofTwoMarkets
may make the difference between a tornado occurring or not occur-
ring in another part of the world.”^5
Picture an empty hockey rink. A person places a puck at midice
and then rolls it toward the far end of the rink. He measures the
angle of his hit, the angle of the puck’s impact, and the angle of its
rebound. He keeps measuring the puck’s angles of impact and re-
bound as the puc kbounces around the rin k.
Assuming no friction, a rule of puc kmotion in the rin kis that
it will emerge from impacting a side of the rin kat precisely its angle
of approach (a similar thing will be familiar to anyone who has
played billiards). That rule means we have defined a deterministic
system under which the future position of the puc kat any time can
be forecast perfectly (assuming its actual or average speed is known).
But now assume that the initial position of the puc kis varied by
a few degrees, even an infinitesimally small variation, not observed
by or known to the forecaster. The forecaster’s predictions of the
puck’s location after it hits the first one or two sides may be impre-
cise—off by some small amount—but the imprecision will be neg-
ligible. The amount of error will grow exponentially, however, with
each subsequent impact. In a short time the forecast will be wide of
the mark.
Disturbing the measure of the puck’s initial position causes its
movement to appear random and unpredictable, whereas knowing
that measure enables precise prediction. It is this sensitive depen-
dence on initial conditions that is the signal characteristic of chaotic
systems.^6 To detect its presence, Lorenz and his followers developed
a couple of fascinating tools.
Pictures and Attractors
Time-series data are conventionally plotted using simple Cartesian
geometry. For example, to plot a time series of a stock’s price, price
is plotted on the vertical axis and chronological time is plotted on
the horizontal axis.
In physics, the usual Cartesian graphs can be turned into more
powerful pictures called phase portraits plotted in phase space, a
presentation that can depict the full range of possibilities for a sys-
tem. The pendulum is the paradigm for illustrating the differences
between Cartesian plots of time-series data and phase portraits of
the same data as well as for introducing the notion of the attractor.
Consider a regularly swinging pendulum driven by mechanical