Global Dynamical Equations and Lyapunov Exponents From Noisy Chaotic Time Series

Abstract

We discuss the extraction of few-parameter, global dynamical models from noisy time series of chaotic systems. In particular, we consider the class of models which are approximations to sets of dynamical equations in the reconstructed phase space. We show that certain numerical methods significantly improve the quality of the resulting models, and central to these methods is the idea of eliminating model terms which are “dynamically insignificant” and add only numerical noise. For the purposes of the paper, we quantify model quality by the rather strict measure of its ability to recover the dynamical invariants of the original system, in particular, the Lyapunov spectrum. Consequently, we also postulate that by first extracting a global model, the Lyapunov spectrum of a generating system can be recovered from time series whose noise levels are much higher than current algorithms would allow. We present several numerical examples to demonstrate the above ideas.

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