Chaos Theory

Chaos Theory

BIBLIOGRAPHY

Chaos theory is a theory of systems dynamics; that is, the analysis of the laws of motion of various systems over time. Complex dynamics investigates why certain systems, while evolving in a predictable fashion for some time, may display at other times a behavior, which looks erratic (random), hence unpredictable. In this context, the main thrust of chaos theory has been to demonstrate that behind this seemingly random behavior or disorder, there is, however, a deterministic underlying structure, which can be described and analyzed by means of differential equations that do not involve uncertainty.

Formally chaos is a nonlinear deterministic process that looks random, a case in which a dynamic mechanism yields a time path so erratic that it passes most standard statistical tests of randomness. Chaotic time paths often have the following features: (1) a trajectory that sometimes displays sharp qualitative changes, such as those associated with large random disturbances; (2) a time path that is extremely sensitive to microscopic changes in the values of its parameters; and (3) a time path that never returns to any point it had previously traversed, but which may, however, display an oscillatory pattern in a certain bounded region. The terms chaos, strange attractors, and complex dynamics have been used interchangeably in the literature to characterize these complex processes.

In linear dynamics, small causes give rise to small effects, and large causes produce large effects. Hence there is a certain sense of proportionality in linear thinking. Nonlinearity, on the other hand, connotes lack of proportionality: very small causes (small changes in the initial conditions, for instance) can give rise to very large effects. One implication of chaos theory, in this context, is to show that nonlinearities are not the exception but the rule of nature and life. Weather forecasting, for instance, is difficult because very small fluctuations in the environment give rise to very large-scale changes. Chaotic systems often possess fractal structures and time-dependent feedback mechanisms. A fractal structure consists of two major features: self-similarity (or scale-invariance) and lack of smoothness. Self-similarity refers to a system that always looks the same regardless of how many times the system is magnified. On the other hand, lack of smoothness relates to the disconnected appearance of fractals.

While the origins of chaos theory date back to the seminal work conducted in 1890 by the French mathematician Jules Henri Poincaré on the so-called three-body problem, it is Edward Lorenz’s 1963 research on atmospheric dynamics and Benoit Mandelbrot’s pathbreaking 1983 investigation of fractal geometry that have rekindled interest in chaos theory since the mid-twentieth century. Applications of complex dynamics have found fertile grounds in several fields such as fluid dynamics, plasma physics, chemistry, electrical engineering, signal processing, cardiology, finance, and time series econometrics and economics, some of which are surveyed in Julian Sprott’s Chaos and Time Series Analysis (2003).

The roots of chaos analysis in economics can be traced back to the literature on business cycles, that is, the analysis of irregular fluctuations in the output level of an economy. While exogenous business cycle theories have become the orthodoxy during the last four decades, the emergence of chaos theory hinted at the possibility that erratic output fluctuations are due to the complex interaction of economic factors, and hence may be endogenously generated.

Chaos studies in theoretical economics have attempted to model economic systems in such a way that chaotic dynamics emerge during the adjustment period to equilibrium or in the evolution of the system itself over time. On the other hand, several empirical investigations have been done in order to identify chaotic behavior in financial and economic time-series data. While earlier empirical studies claimed to have found evidence of chaotic behavior in a number of economic time series, such as U.S. business cycle data, various monetary aggregates, and precious metal prices, subsequent research, such as that conducted by Aydin Cecen and Cahit Erkal in 1996, demonstrated that there is little evidence in favor of deterministic chaos in exchange rate returns.

SEE ALSO Catastrophe Theory; Shocks

BIBLIOGRAPHY

Cecen, Aydin A., and Cahit Erkal. 1996. Distinguishing Between Stochastic and Deterministic Behavior in Foreign Exchange Rate Returns: Further Evidence. Economics Letters 51: 323-329.

Lorenz, Edward. 1963. Deterministic Nonperiodic Flow. Journal of the Atmospheric Sciences 20: 130-141.

Mandelbrot, Benoit B. 1983. Fractal Geometry of Nature. New York: W. H. Freeman.

Poincaré, Jules Henri. 1890. Sur le Problème de Trois Corps et les Equations de la Dynamique. Acta Mathematica 13: 1-270.

Sprott, Julian C. 2003. Chaos and Time Series Analysis. New York: Oxford University Press.

Aydin A. Cecen