ChaoticMarket 37publication of a volume in 1988. It has been carried into the realm
of “phynance”—the merger of physics and finance—most spectac-
ularly by Doyne Farmer and Norman Packard, whose phynance ex-
ploits are chronicled by Thomas Bass in his 1999 bookThe Predic-
tors. As a result of these works, chaos theory became an important
and growing field in the study of the nonlinear dynamic behavior of
economic and financial systems.
Through chaos theory, physicists discovered that many phenom-
ena in the universe previously thought to be random (unpredictable,
exhibiting no pattern) are not random but exhibit a significant pat-
tern. To oversimplify, chaos theory holds that there is a pattern to
the seeming randomness of physical events occurring in the uni-
verse. Thus, systems that appear to be stochastic (to involve only
random motion or behavior under conventional linear modeling)
may be deterministic, or exhibit more complex internal dependence
than simple linear modeling reveals.
Chaos theory has its roots in the nineteenth-century wor kof
Henri Poincare ́, a French mathematician and physicist who studied
the famous three-body problem.^4 Newton, using his laws of motion
and gravitation, proved that it was possible to calculate accurately
the future positions and velocities of two mutually attractive material
bodies. Neither Newton nor anyone since, however, has been able
to do so for three or more bodies.
This three-body problem reveals itself repeatedly to scientists
sending space probes to Mars and other planets: They chart a course
directed to where the planet will be in its orbit when the probe
arrives (not where the planet is upon sending the probe), but mid-
course corrections are nevertheless necessary because Newtonian
physics can predict accurately only the interaction of two bodies, not
three. (This has led some probes to be lost in space.)
Poincare ́ attributed the three-body problem to nonlinearities in-
herent in multibody systems as the result of which “small differences
in the initial conditions produce very great ones in the final phe-
nomena. A small error in the former will produce an enormous error
in the latter.” This insight, now the unifying core of chaos theory, is
known as “sensitive dependence upon initial conditions.”
The classic example of sensitivity to initial conditions is the but-
terfly effect in meteorology. Its pioneer in the early 1960s was the
MIT meteorologist Edward Lorenz, who said, “The dynamical equa-
tions governing the weather are so sensitive to the initial data that
whether or not a butterfly flaps its wings in one part of the world