By L. Douglas Kiel, Euel W. Elliott
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Additional resources for Chaos Theory in the Social Sciences: Foundations and Applications
Sample text
As a matter o f fact, all models o f dynamical systems w i l l contain some error. One source o f error is the computer. For many models, the only source o f error is that due to computer rounding error. I n physical experiments, error can creep into the measurements from imprecision in the instruments used to take mea surements. Social scientists are well aware that many measurements we make are imprecise. Inferences from surveys rely on sampling theory and the sam ple error increases as the sample size decreases.
Such systems are potentially chaotic because their initial conditions can be fixed only w i t h finite accuracy. Error may explode. Yet the long-time unpredictability exists only at the individual level (or, in the language o f dynamics, at the level o f individual trajectories). A t the level o f statistical properties o f the time evolution (averaged over different trajectories, say as they evolve from different nearby initial conditions), very definite predictions are possible. Said differently, when or where chaos exists can only be k n o w n by statistical prediction.
L i m i t cycles occur when the orbit is drawn toward a cyclic path, rather than toward a point fixed in space. Attractors can also be complex in that they may be quasi-periodic or strange. A n example o f a quasi-periodic attractor is an orbit on a torus, a doughnut generated by the cosines o f a pair o f incom mensurate frequencies. Chaos Characteristics Chaos is measured by the characteristics o f its dynamics: fractal structure o f the attractor; metric K entropy; and the Lyapunov characteristic exponent, A,.