Chaos Theory

Chaos theory refers to a type of behavior with nonlinear dynamics that is both irregular and oscillatory. It is encountered mathematically with certain sets of nonlinear deterministic dynamic models in which patterns of overtime behavior are not repeated no matter how long the model continues to operate. Some discrete-time models using nonlinear difference equations are known to exhibit chaotic dynamics under certain conditions using only one equation. However, in continuous-time models, the possibility of chaos normally requires a minimum of three independent variables, which usually requires an interdependent system of three differential equations. This requirement for continuous-time models can be changed in special cases if the system also has a forced oscillator or time lags, in which case chaos can appear even in single-equation continuous-time models.

Chaos can occur only in nonlinear situations. In multidimensional settings, this means that at least one term in one equation must be nonlinear while also involving several of the variables. Because most nonlinear models (and nearly all of the substantively interesting ones) have no analytical solutions, they must be investigated using numerically intensive methods that require computers. Because the dimensionality and nonlinearity requirements of chaos do not guarantee its appearance, chaotic behavior is typically discovered in a model through computational experimentation that involves finding variable ranges and parameter values that cause a model to display chaotic properties.

Chaotic processes may be quite common in real physical and even social systems, even though our present ability to identify and model such processes is still developing. Discovering real chaotic processes in physical and social systems is often quite difficult because stochastic noise is nearly always present as well in such systems, and it is not easy to separate truly random behavior from chaotic behavior. Nonetheless, mathematical tools continue to be developed that are aimed at sorting out these processes and issues.

Chaos has three fundamental characteristics. They are (a) irregular periodicity, (b) sensitivity to initial conditions, and (c) a lack of predictability. These characteristics interact within any one chaotic setting to produce highly complex nonlinear variable trajectories. Irregular periodicity refers to the absence of a repeated pattern in the oscillatory movements of the chaotically driven variables. Because of the irregular periodicity, Fourier analysis, graphing techniques, and other methods are commonly used to build a case for identifying chaotic processes (see Brown, 1995a).

The nonlinear model that has been among the most well studied with regard to chaos in discrete settings is a general form of a logistic map, and its chaotic properties were initially investigated by May (1976). This general logistic map is Yt+1 = aYt (1−Yt). Under [Page 121]the right conditions, this map can produce the standard S-shaped trajectory that is characteristic of all logistic structures. However, oscillations in the trajectory occur when the value of the parameter a is sufficiently large. For example, when the value of parameter a is set to 2.8, the trajectory of the model oscillates around the equilibrium value of Yt+1 = Yt = Y * while it converges asymptotically toward this equilibrium limit. But when the value of the parameter a is set equal to, say, 4.0, the resulting longitudinal trajectory never settles down toward the equilibrium limit and instead continues to oscillate irregularly around the equilibrium in what seems to be a random manner that is caused by a DETERMINISTIC process.

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