Investigating the use of manifold embedding for attractor reconstruction from time series

Abstract

Spatio-temporal analysis of a time series from a complex dynamical system often requires reconstruction of the state-space attractor from observations of a single state variable. The standard approach takes advantage of the Takens delay embedding theorem to obtain the reconstruction. We investigate here a modification which makes use of nonlinear spectral graph techniques for learning the underlying manifold from high-dimensional data. Specifically, we examine how well diffusion maps and locally-linear embedding recover system dynamics and their sensitivity to parameters. Analysis is conducted using individual observations of the chaotic Lorenz and Hénon attractors. We show that manifold embeddings, given selected parameter choices, can improve forecasting capability for chaotic time series.