nonlinear dynamics


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nonlinear 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 theorychaos 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.
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). Unlike a linear system, in which a small change in one variable produces a small and easily quantifiable systematic change, a nonlinear system exhibits a sensitive dependence on initial conditions: small or virtually unmeasurable differences in initial conditions can lead to wildly differing outcomes. This sensitive dependence is sometimes referred to as the "butterfly effect," the assertion that the beating of a butterfly's wings in Brazil can eventually cause a tornado in Texas. Historically, in fact, one of the first nonlinear systems to be studied was the weather, which in the 1960s 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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 sought to model by a relatively simple set of equations. He discovered that the outcome of his model showed an acute dependence on initial conditions. Later work revealed that underlying such chaotic behavior are complex but often aesthetically pleasing geometric forms called strange attractors. Strange attractors exist in an imaginary space called phase space, in which the ordinary dimensions of real space are supplemented by additional dimensions for the momentummomentum
, in mechanics, the quantity of motion of a body, specifically the product of the mass of the body and its velocity. Momentum is a vector quantity; i.e., it has both a magnitude and a direction, the direction being the same as that of the velocity vector.
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 of the system under investigation. A strange attractor is a fractal, an object that exhibits self-similarity on all scales. A coastline, for instance, looks much the same up close or far away. Nonlinear dynamics has shown that even systems governed by simple equations can exhibit complex behavior. The evolution of nonlinear dynamics was made possible by the application of high-speed computers, particularly in the area of computer graphics, to innovative mathematical theories developed during the first half of the 20th cent. Three branches of study are recognized: classical systems in which friction and other dissipative forces are paramount, such as turbulent flow in a liquid or gas; classical systems in which dissipative forces can be neglected, such as charged particles in a particle acceleratorparticle accelerator,
apparatus used in nuclear physics to produce beams of energetic charged particles and to direct them against various targets. Such machines, popularly called atom smashers, are needed to observe objects as small as the atomic nucleus in studies of its
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; and quantum systems, such as molecules in a strong electromagnetic field. The tools of nonlinear dynamics have been used in attempts to better understand irregularity in such diverse areas as dripping faucets, population growth, the beating heart, and the economy.

Bibliography

See S. N. Rasband, Chaotic Dynamics of Nonlinear Systems (1990); A. J. Lichtenberg and M. A. Lieberman, Regular and Chaotic Dynamics (1992); S. J. Guastello, Chaos, Catastrophe, and Human Affairs: Applications of Nonlinear Dynamics to Work Organizations and Social Evolution (1995); A. H. Nayfeh and B. Balachandran, Applied Nonlinear Dynamics: Analytical, Computational, and Experimental Methods (1995).

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