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).

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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.

Collins Dictionary of Astronomy © Market House Books Ltd, 2006

chaos theory

a theory, applied in various branches of science, that apparently random phenomena have underlying order
Collins Discovery Encyclopedia, 1st edition © HarperCollins Publishers 2005
References in periodicals archive ?
Liu, "Colour image encryption based on advanced encryption standard algorithm with two-dimensional chaotic map," IET information security, 7(4), pp.
For this reason, chaotic maps should be used instead of random number sequences to improve the performance of the WOA.
In this work, attractive movement of fireflies was simulated with ten chaotic maps. Chaotic sequences are used for parameters a and A in the Chaotic GWO approach [27].
Since the sidelobe of the chaotic map is zero, (14) can be simplified to
Chaotic maps have the properties of unpredictability and sensitivity to their parameters and initial values.
Conventionally, three area-preserving invertible chaotic maps, that is, the cat map, the baker map, and the standard map, are widely used for image scrambling.
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.
As observed from this figure, the green component of original color image is first multiplied by the first chaotic map: the proposed chaotic map, and this stage is followed by multiplying by the second chaotic map: the proposed map with different initial conditions.
Different chaotic maps are introduced to enhance the effectiveness and robustness of the algorithm.
In the proposed scheme, a certain number of users are grouped together that collaborate by sharing orthogonal chaotic-spreading sequences generated by 1D chaotic map. For example, using chaotic sequences of length 31, the proposed scheme supports 120 full-rate users compared to 90 and 30 users supported by collaborative CDMA and the conventional CDMA, respectively.
Bifurcation diagram plots output sequences of a chaotic map along with the change of its system parameter(s).
Keywords: Prime gaps, invariant density, chaotic map, inverse Frobenius-perron