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  Search   1 : CHAOS                                           57 Documents
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 Entry 6 of 57                                                   Jun 8, 1992 

Chaos {kay'-ahs}

     In Greek mythology, Chaos was the unorganized state, or void,
     from which all things arose. Proceeding from time, Chaos
     eventually formed a huge egg from which there issued Heaven,
     Earth, and Eros (love). According to Hesiod's Theogony, Chaos
     preceded the origin not only of the world, but also of the
     gods.



chaos theory

     Chaos theory, a modern development in mathematics and science,
     provides a framework for understanding irregular or erratic
     fluctuations in nature.  Chaotic systems are found in many
     fields of science and engineering.  Evidence of chaos, for
     example, has been found in models and experiments describing
     convection and mixing in fluids, wave motion, oscillating
     chemical reactions, and electrical currents in semiconductors.
     It is also found in the dynamics of animal populations and of
     medical disorders.  In addition, attempts are being made to
     apply chaotic dynamics to phenomena observed in the social
     sciences.

     A chaotic system is defined as one that shows "sensitivity to
     initial conditions." That is, any uncertainty in the initial
     state of the given system, no matter how small, will lead to
     rapidly growing errors in any effort to predict the future
     behavior.  For example, the motion of a dust particle floating
     on the surface of a pair of oscillating whirlpools can display
     chaotic behavior.  The particle will move in well-defined
     circles around the centers of the whirlpools, alternating
     between the two in an irregular manner.  An observer who wants
     to predict the motion of this particle will have to measure its
     initial location.  If the measurement is not infinitely
     precise, however, the observer will instead obtain the location
     of an imaginary particle very close by.  The "sensitivity to
     initial conditions" mentioned above will cause the nearby
     imaginary particle to follow a path that diverges from the path
     of the real particle.  This makes any long-term prediction of
     the trajectory of the real particle impossible.  In other
     words, the system is a chaotic one.

     The possibility of chaos in a natural, or deterministic, system
     was first envisaged by the French mathematician Henri Poincare
     in the late 19th century, in his work on planetary orbits.  For
     many decades thereafter, however, little interest was shown in
     such possibilities.  The modern study of chaotic dynamics may
     be said to have begun in 1963, when American meteorologist
     Edward Lorenz demonstrated that a simple, deterministic model
     of thermal convection in the Earth's atmosphere showed
     sensitivity to initial conditions--or, in current terms, that
     it was a chaotic system.

     Following this observation, scientists and mathematicians began
     to study the progression from order to chaos in various
     systems, as the parameters of the systems were varied.  In 1971
     a Belgian physicist, David Ruelle, and a Dutch mathematician,
     Floris Takens, together predicted that the transition to
     chaotic turbulence in a moving fluid would take place at a
     well-defined critical value of the fluid's velocity (or some
     other important factor controlling the fluid's behavior).  They
     predicted that this transition to turbulence would occur after
     the system had developed oscillations with at least three
     distinct frequencies.  Experiments with rotating fluid flows
     conducted by American physicists Jerry Gollub and Harry Swinney
     in the mid-1970s supported these predictions.

     Another American physicist, Mitchell Feigenbaum, then predicted
     that at the critical point when an ordered system begins to
     break down into chaos, a consistent sequence of period-doubling
     transitions would be observed.  This so-called "period-doubling
     route to chaos" was thereafter observed experimentally by
     various investigators, including the French physicist Albert
     Libchaber and his coworkers.  Feigenbaum went on to calculate a
     numerical constant that governs the doubling process
     (Feigenbaum's number) and showed that his results were
     applicable to a wide range of chaotic systems.  In fact, an
     infinite number of possible routes to chaos can be described,
     several of which are "universal," or broadly applicable, in the
     sense of obeying proportionality laws that do not depend on
     details of the physical system.

     The term chaotic dynamics refers only to the evolution of a
     system in time.  Chaotic systems, however, also often display
     spatial disorder--for example, in complicated fluid flows.
     Incorporating spatial patterns into theories of chaotic
     dynamics is an active area of current research.  Ultimately,
     scientists and mathematicians hope to extend theories of chaos
     to the regime of fully developed turbulence, where complete
     disorder exists in both space and time.  This effort is widely
     viewed as among the greatest challenges of modern physics.
     JERRY GOLLUB and THOMAS SOLOMON



 Cited References 

Bibliography:  Drutchfield, J.  P., et al., "Chaos," Scientific
American, December 1986;  Gleick, James, Chaos:  Making a New
Science (1987);  Kadanoff, L.P., "Routes to Chaos," Physics
Today, December 1983.