Smoothing data with local instabilities for the identification of chaotic systems

Abstract

Chaotic systems are characterized by the local divergence of nearby orbits in state-space. This provokes a sensitive dependence on initial conditions and, in turn, drastically limits the accuracy of long-term predictions. This has important implications in the filtering of data generated by such systems. In the present paper, the use of global smoothers for chaotic data is investigated. The ultimate objective is to be able to identify dynamically valid models from smoothed data when the identification from the original noisy data has completely failed. The objective is to produce identified models that faithfully reproduce the dynamical invariants of the original system, such as the geometry of the attractors in state-space, the largest Lyapunov exponent, fractal dimensions and Poincaré sections. Numerical examples are included, which illustrate the main points of the paper.