Chaos Theory

Chaos theory describes the motion of certain dynamical, nonlinear systems under specific conditions. Chaotic motion is not the same as random motion. It is especially likely to emerge in systems that are described by at least three nonlinear equations, though it may also arise in other settings under specific conditions. All of these systems are characterized by a sensitivity to initial conditions within bounded parameters. Chaotic systems must also be transitive (any transformation in period t1 will continue and overlap in period t2), and its periodic orbits are dense (for any point in the system y, there is another point with a distance d = y in the same periodic orbit).

The history of this branch of study is a complex, interdisciplinary affair, with scholars in different fields working on related problems in isolation, often unaware of research that had gone before. One of the most important characteristics of chaotic systems is their sensitivity to initial conditions. In 1961, one of the fathers of chaos theory,Edward Lorenz, accidentally discovered this principle while studying a simple model of weather systems constructed from no more than 12 parameters. Wishing to review a certain set of results, he manually reentered values for these parameters from a printout and started the simulation again in midcourse. However, the new set of predictions that the computer made were vastly different from the first series that had been generated. After ruling out mechanical failure, Lorenz discovered that that by reentering the starting values of the parameters, he had truncated the decimals from five places to three. Lorenz and his colleges had assumed small variance in the inputs of a set of equations would lead to a likewise small variance in the outcomes. Yet in a system of complex or chaotic movement, very small variance in initial conditions can lead to large difference in outcomes. This property is popularly referred to as the “butterfly effect.”