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 ?
The logistic map is a well-known polynomial chaotic map which can be described as:
Mathematically, it is already established that when [alpha] = 4, then all the values generated by the chaotic map will lie in the range of 0 -1 except that [cm.
Kawamura, Chaotic maps on a measure space and the behavior of the orbit of a state, Tokyo Jour.
Pseudo-random number generator based on mixing of three chaotic maps.
Step 2: initializing HM by iterating the selected chaotic maps until if reaches the HMS.
In this algorithm PAR and bw values have not been fixed in HS and they have been modified by the selected chaotic maps as follows
A secret key cryptosystem by iterating a chaotic map.
Again there are number of ways through which digital chaos-based cryptosystems be realized: block ciphers based on forward and/or reverse iterations of chaotic maps [4, 20-23], block ciphers based on chaotic round functions [24-27], stream ciphers implementing chaos-based pseudo random bit generators (PRBG) [28-33] etc.
13] have proposed a symmetric image encryption in which a two-dimensional chaotic map is generalized to three-dimension for designing a real time secure image encryption scheme.
Understanding the forbidden patterns of chaotic maps is important because the absence of these patterns is what distinguishes sequences generated by chaotic maps from random sequences.
In general, chaotic maps are obtained by a deterministic rule: [x.
Some topics addressed include securing XML with role-based access control in the health care environment, a mobile-based aggregation architecture for user-centric identity management, a new encryption algorithm based on chaotic maps for wireless sensor networks, and a polling booth-based electronic voting scheme.