chaos theory

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chaos theory,

in mathematics, physics, and other fields, a set of ideas that attempts to reveal structure in aperiodic, unpredictable dynamic systems such as cloud formation or the fluctuation of biological populations. Although chaotic systems obey certain rules that can be described by mathematical equations, chaos theory shows the difficulty of predicting their long-range behavior. In the last half of the 20th cent., theorists in various scientific disciplines began to believe that the type of linear analysis used in classical applied mathematics presumes an orderly periodicity that rarely occurs in nature; in the quest to discover regularities, disorder had been ignored. Thus, chaos theorists have set about constructing deterministic, nonlinear dynamic models that elucidate irregular, unpredictable behavior (see nonlinear dynamicsnonlinear dynamics,
study of systems governed by equations in which a small change in one variable can induce a large systematic change; the discipline is more popularly known as chaos (see chaos theory).
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). Some of the early investigators of chaos were the American physicist Mitchell Feigenbaum; the Polish-born mathematician and inventor of fractals (see fractal geometryfractal geometry,
branch of mathematics concerned with irregular patterns made of parts that are in some way similar to the whole, e.g., twigs and tree branches, a property called self-similarity or self-symmetry.
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) Benoit MandelbrotMandelbrot, Benoît B.
, 1924–2010, French-American mathematician, b. Warsaw, Poland, Ph.D. Univ. of Paris, 1952. Largely self-taught and considered a maverick in the field of mathematics, he was uncomfortable with the rigorously pure logical analysis prescribed by
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; the American mathematician James Yorke, who popularized the term "chaos"; and the American meteorologist Edward LorenzLorenz, Edward Norton,
1917–2008, American meteorologist and pioneer of chaos theory, b. West Hartford, Conn., Ph.D. Massachusetts Institute of Technology, 1948. Lorenz became interested in meteorology while working as a weather forecaster during World War II, and after
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See J. Gleick, Chaos: Making a New Science (1987); I. Stewart, Does God Play Dice?: The Mathematics of Chaos (1989); A. A. Tsonis, Chaos: From Theory to Applications (1992); D. N. Chorafas, Chaos Theory in the Financial Markets (1994).

chaos theory

The theory of the unpredictable behavior that can arise in systems obeying deterministic scientific laws – laws that under ideal conditions completely determine the future states of a system from its preceding states. In practice, however, quantities cannot be measured with unlimited precision and the predictability suffers as a result of input errors. In a typical nonchaotic system, the errors accumulate with time but remain managable. In a chaotic system, there is a sensitivity to variations in the initial conditions. Input errors are multiplied at an escalating rate until all predictive power is lost and the system behaves in an apparently random manner.

There are many apparently simple physical systems in the Universe that obey deterministic laws but yet behave unpredictably.

chaos theory

a theory, applied in various branches of science, that apparently random phenomena have underlying order
References in periodicals archive ?
Although at first it may seem counterintuitive, the concept of deterministic chaos may be found in countless natural phenomena, from a dripping faucet to species populations.
The relevant features of the primary domain - that is, of deterministic chaos - an be illustrated with reference to turbulence, which may be defined as the motion of a fluid in which local velocities and pressures fluctuate randomly (my own definition, dredged up from engineering courses years ago).
Any system displaying deterministic chaos will also be characterized by "sensitive dependence on initial conditions.
Consequently, even though the driving influence of deterministic chaos is detected in the above three currencies, predictability of these exchange rates is highly improbable as this deterministic influence is high-dimensional.
Understanding the Meaning of Nonlinear Dynamics and Deterministic Chaos
We might be able to identify clinical situations that show evidence of deterministic chaos and that might therefore be better candidates for brief and apparently minor interventions that can bring forth transformative results.
In some cases, particularly when a system is in a state of deterministic chaos, a very small intervention can have a very large effect.
The purpose of this article was to empirically investigate the possibility that the time series of insurance stock returns exhibit nonlinear structure and low deterministic chaos.
The CD is used to differentiate between deterministic chaos and stochastic systems.
However, additional tests are needed to discriminate between nonlinearities due to stochastic behavior and nonlinearities due to the existence of deterministic chaos.
Second, the interest of this study lies only in detecting low dimensional deterministic chaos.
However, in the absence of identification of a chaotic attractor, very possibly an artifact of data limitations, one is unable to obtain conclusive proof of the existence of deterministic chaos.

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