Deterministic chaos, often just called "chaos", refers in
the world of dynamics to the generation of random, unpredictable behavior
from a simple, but nonlinear rule. The rule has no "noise", randomness,
or probabilities built in. Instead, through the rule's repeated application
the long-term behavior becomes quite complicated. In this sense, the unpredictability
"emerges" over time.
There are a number of characteristics one observes in a deterministically
Long term behavior is difficult or impossible to predict: Even very
accurate measurements of the current state of a chaotic system become useless
indicators of where the system will be. One has to measure the system again
to find out where it is.
Sensitive dependence on initial conditions (a property noted by
Starting from very close initial conditions
a chaotic system very rapidly moves to different states.
Broadband frequency spectrun: That is, the output from a chaotic system
sounds "noisy" to the ear. Many frequencies are excited.
Exponential amplification of errors: In any real world setting small
amounts of external noise rapidly grow to control the sytem. If this noise
is below measurement accuracy, so that an experimenter can't see or control
the noise, then the system appears unpredictable. The microscopic "heat
bath" is amplified to human scales.
Local instability versus global stability: In order to have amplification
of small errors and noise, the behavior must be locally unstable: over
short times nearby states move away from each other. But for the system
to consistently produce stable behavior, over long times the set of behaviors
must fall back into itself. The tension of these two properties leads to
very elegantly structured chaotic attractors.
Professor James Yorke, an applied mathematician at the University of
Maryland, is often credited with associating the word "chaos"
with these particular mechanisms, in the late 1970s. While it has helped
the field of nonlinear dynamics to have a simple, handy word like "chaos",
the word itself is a bit of a misnomer. In fact, the word is downright
confusing, if one interprets it in the nontechnical sense of common language--"lack
of order". In fact, deterministic chaotic systems are quite ordered
and even predictable on short time scales. In many ways modern dynamicists
study deterministic chaotic systems to understand the interplay between
order and "utter chaos". The goal is to find the hidden order
in the apparent chaos.
One of the earlist known experimental reports of deterministic chaos
occurred during 1927 in the Britsh scientific journal Science. A brief
letter to the editor, only two pages long, by the Dutch electrical engineer
Balthasar van der Pol
and his colleague van der Mark, reports on the "irregular
noise" heard in a telephone earpiece attached to an electronic
tube circuit. Unfortunately for the authors, the paper discounts this phenomenon
and instead the paper concentrates on various periodic, predictable behaviors
that sounded like "bag-pipes" and were found in other non-chaotic
regimes of the circuit.
Today deterministic chaotic behavior has been discovered in numerous
natural phenomena and analyzed in detail in dozens of experiments. From
to dripping faucets,
from predator-prey ecologies to measle
epidemics, from oscillating chemical reactions to irregular beats of a
chicken heart, the underlying mechanisms have been detected. Despite the
scientific successes, though, it is important to emphasize that deterministic
chaos, and the various mechanisms that underlie it, are not the only explanations
of random, noisy, unpredictable behavior in nature. Many well-known processes,
and undoubtedly many waiting to be discovered, can produce behavior that
is unpredictable. Thus, this abiding question is, How do we discover which
of many possible mechanisms has produced the apparent disorder?