A quasi-diagonal approach to the estimation of Lyapunov spectra for spatio-temporal systems from multivariate time series

We describe methods of estimating the entire Lyapunov spectrum of a spatially extended system from multivariate time-series observations. Provided that the coupling in the system is short range, the Jacobian has a banded structure and can be estimated using spatially localised reconstructions in low embedding dimensions. This circumvents the ``curse of dimensionality'' that prevents the accurate reconstruction of high-dimensional dynamics from observed time series. The technique is illustrated using coupled map lattices as prototype models for spatio-temporal chaos and is found to work even when the coupling is not strictly local but only exponentially decaying.Comment: 13 pages, LaTeX (RevTeX), 13 Postscript figs, to be submitted to Phys.Rev.

Estimation of Delay Times From a Delayed Optical Feedback Laser Experiment

. -- We estimate delay times from high-dimensional chaotic time series experimentally obtained from a fast optical time-delayed feedback system. The experiment consists of a semiconductor laser, where the instabilities are induced by an external T-shaped cavity introducing two delay times into the laser. The delay times are determined by a filling factor analysis and found to give a better estimate than those obtained by autocorrelation functions. Finally, the possibility of this method for the reconstruction of the system's di#erential equations is discussed. The characterization of nonlinear dynamical behavior and the identification of the underlying deterministic time-evolution laws from experimental time series has turned out to be one of the key problems in the study of nonlinear dynamical systems. For dynamical systems with a low number of degrees of freedom, embedding techniques [1] have been exceptionally successful for the computation of chaotic indicators (dimensions, Lyapun..