Largest Lyapunov-exponent estimation and selective prediction
by means of simplex forecast algorithms.

R.M. Dünki, Physical Review E 62: 6505 - 6515, (2000)

Limited predictability is one of the remarkable features of deterministic chaos and this feature may be quantized in terms of Lyapunov-exponents. Accordingly, Lyapunov-exponent estimates may be expected to follow in a natural way from forecast algorithms. Exploring this idea, we propose a method estimating the largest Lyapunov-exponent from a time series which uses the behaviour of so-called simplex forecasts. The method considers estimation of properties of the distribution of local simplex expansion coefficients. These are also used for the definition of error bars for the Lyapunov-exponent estimates and allows for selective forecasts with improved prediction accuracy. We demonstrate these concepts on standard test examples and three realistic applications to time series concerning largest Lyapunov-exponent estimation of an experimentally obtained hyperchaotic NMR-signal, brain state differentiation and stock market prediction.