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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.

Bibliography

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 ?
065, the gear system will be from periodic 1 motion to chaotic motion.
Based on the Melnikov method and its extensions to resonance cases developed by Kovacic and Wiggins, the thin-walled compressor blade can undergo homoclinic bifurcation and the Shilnikov-type homoclinic orbit; that is, there exists chaotic motion in full four-dimensional averaged system.
Through three sets of bifurcation diagrams, we find that breathing crack plate has complex nonlinear phenomena such as quasi-periodic motion, bifurcation motion, and chaotic motion, and the nonlinear dynamic behaviors of open-crack model are similar to intact plate model, which merely presents the single-cycle motion and the double periodic motion.
Other areas described include characteristics of chaotic motion based on the wavelet ridge, a second derivative Gaussian algorithm in engine fault simulation, free vibration of a CLD combination cylindrical shell, and modal analysis of a five-layer vertical structure platform.
The diagrams in Figs 5 - 6 and the video clip (Physical double pendulum, 2011) show the chaotic motion of double pendulum.
Some specific subjects covered include chaotic motion of relativistic electrons driven by Whistler waves, thermodynamic and transport properties of non-ideal complex plasma, and linear coupling of electron cyclotron waves in magnetized plasmas.
This means, for the intersecting vortex field of the three vortices, the effect of translational superposition changes the fixed point behavior into an interesting chaotic motion.
While Alex Turner and his bandmates - with the exception of furious drummer Matt Helder - work their way through song after song, their fans are an ever-exuberant mass of chaotic motion.
Following a logical profession from the lower part of the body (leg compressions), up to the arm and wrist (finger compressions), and ending at the neck (choke holds), Petrillie goes on to teach the viewer how to access these locks from both the chaotic motion of a fight and from a standing position after an opponent has been softened up through some strikes.
Examples of chaotic motion in the solar system include the rotation of the Saturnian satellite Hyperion, the orbital evolution of numerous asteroids and comets, and the orbit of Pluto.
Under the intoxicating influence of nonlinearity, only a few degrees of freedom are necessary to generate chaotic motion.