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.