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 ?
High-dimensional chaotic map has a minimum of two parameters that form a complicated structure, and it shows good chaotic performance.
The proposed system is used for resisting the guessing password attacks by using the Chaotic Map Discrete Logarithm problem (CMDLP) and Chaotic Map Computational Diffie-Hellman problem (CMCDHP) and enhancing the privacy protection.
Chaotic signal can be readily obtained from a chaotic oscillator, a chaotic map.
Finally, in this paper merely logistic map has been embedded to diversify the population of CTLBO algorithm; however, other different chaotic maps will be analyzed in the future work.
This algorithm is the combination of the chaotic map and charge system search algorithm.
However, the sidelobes of some radar signals generated from chaotic map are not so good, so they are not suitable for high resolution radar imaging [9, 10].
Chaos is understood as the complex, bounded and unstable behavior caused by a simple deterministic nonlinear system or chaotic map, such that, the generated sequences are quasi-random, irregular, ergodic, semi-stochastic and very sensible to the initial value [30].
Sasaki, Banach lattices and the Perron- Frobenius operator associated with chaotic map, Far East Journal of Dynamical Systems, 81(2006), 1-25.
The Baker map is a chaotic map that generates a permuted version of a square matrix.
Figure 1 also illustrates the property of period folding of trajectories in chaotic systems, and demonstrates the concept of low dimension: the chaotic map of [x.
Azaceta has rendered this darkly comic scenario even more oppressive by boxing in the refugee with overlapping pattern of lines, grids, and circles that on the one hand delineate a shallow, claustrophobic, representational space, and on the other a flat, chaotic map.