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 ?
Chaotic systems have properties like ergodicity, sensitivity to initial condition, mixing property, and also have deterministic dynamic and complex structure.
It has been shown by some experiments that continuous time chaotic systems give more successful results than chaotic maps.
This work investigates the chaos synchronization of fractional-order chaotic systems with different structures based on T-S fuzzy systems, where external disturbances in slaves system are considered.
In this paper, we are concerned with the Q-S synchronization of fractional-order discrete-time chaotic systems with different dimensions and non-identical orders.
Karthikeyan, "Anti-synchronization of Lu and Pan chaotic systems by adaptive nonlinear control," European Journal of Scientific Research, vol.
With the gradual development of memristor, some scholars have applied the synchronization control to memristive chaotic systems, which has aroused a new upsurge in people's research.
For the chaotic systems, Lyapunov exponent is not only an important index to distinguish chaotic attractor, but also a quantitative description of the sensitivity about the initial values.
Lorenz mapping is a typical chaotic mapping in chaotic systems, and the system dynamic equation is
In these literatures, only one memristor was applied in an independent circuit, and the dynamic characteristics of memristive chaotic system are related to the initial state of memristor, including unique nonlinear physics phenomenon.
By early 21st century many researchers have announced different chaotic systems such as Chen system [4] Liu system [5], Sundarapandian system [6], Sundarapandian system [7], and Pham system [8].
Pourmahmood Aghababa, "Robust finite-time stabilization of fractional-order chaotic systems based on fractional lyapunov stability theory," Journal of Computational and Nonlinear Dynamics, vol.
This is the problem with chaotic systems. It often only takes a small change in one variable early in the chain to cause a huge change in the endpoint.