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). 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 Tokyo can eventually change the weather in New York City. Historically, in fact, one of the first nonlinear systems to be studied was the weather, which in the 1960s Edward Lorenz 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 momentum 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 accelerator; 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.

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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