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 chaotic system:
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 Poincare, Birkhoff, and even Turing): 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 compound pendula 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?
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