chaos theory

(redirected from Chaotic motion)
Also found in: Dictionary, Medical, Financial.

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).
..... Click the link for more information.
). 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.
..... Click the link for more information.
) 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
..... Click the link for more information.
; 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
..... Click the link for more information.


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 ?
Figure 3 illustrates chaotic motions for the honeycomb sandwich plate.
Figure 6 indicates that the chaotic motion of the system appears when [[OMEGA].sub.1] = 0.8.
Further, chaotic motion is found from medium interdot distance [kappa] = 5 as shown in Figure 6.
Section 5 presents the Lyapunov exponent used to determine whether the system exhibits chaotic motion. A state feedback control technique for controlling chaos in the suspension system is presented in Section 6.
Ghosal, "Chaotic Motion in a Flexible Rotating Beam and Synchronization," Journal of Computational and Nonlinear Dynamics, vol.
When [epsilon] = 2.065, the gear system will be from periodic 1 motion to chaotic motion. Taken [epsilon] = 2.28 and 2.51, the Poincare map of the system is shown in Fig.
The simple zeros of (7) give the critical value of the forcing amplitude ([F.sub.c]), and the first homoclinic bifurcation occurs when we cross the critical value by increasing the forcing amplitude, implying that the chaotic motion occurs in the system.
Entropy values can be used to distinguish between regular motion, chaotic motion, and random motion.
The color-field artists used to paint huge canvases in which different shades placed next to one another appear to vibrate, and I am guessing the artists picked up the idea of color vibrations from the yellows and greens and orange hues of the October leaves, which appear to be in chaotic motion, even when they are entirely still.
Turbulence--the random, chaotic motion best known for jarring air travelers--is ubiquitous on Earth but always occurs in fluids, such as air or water.
Considering aforesaid, developing strategies for controlling chaos phenomenon based on the features of chaotic motion is highly important; therefore, many nonlinear techniques for chaos control were proposed, such as feedback control [1, 2] and sliding mode control [3-5].
The system chaotic motion can be obtained by Lyapunov exponent with different [K.sub.d], as presented in Figure 12.