Functional reconstruction and local prediction of chaotic time series.

Abstract

A formalism for the global polynomial function approximation for the prediction of chaotic time series is introduced. It takes into account the ergodic invariant measure of the dynamical system. The method presented is based on the generalized Fourier expansion with respect to the polynomial system \ensuremath{\Pi} orthonormal to the invariant measure. The expansion coefficients, which are optimal in the ${\mathit{L}}_{2}$ sense with respect to the natural metrics ${\mathrm{\ensuremath{\rho}}}_{\mathit{I}}$, are expressed in terms of the hierarchy of moments scrM and the hierarchy of functional moments \ensuremath{\Gamma}. In this way, any kind of multivariable parameter optimization becomes unnecessary. The presented approach to reconstruction furnishes good results in the global reconstruction and prediction of time series arising from both discrete and continuous dynamical systems.