Equations of Motion from a Data Series

Temporal pattern learn ing, control and predict ion , and chaotic data an alysis sha re a common pr oblem : ded ucing optimal equations of motion from observations of time-dependent behavior . Each desires to obtain models of the physical world from limited information. We describe a method to reconstruct the deterministic portion of the equations of motion directly from a data series. These eq uations of motion represent a vast reduction of a chaotic data set's observed com plexity to a compact, algorit hm ic specification . T his approach employs an informational measure of model optima lity to gu ide searching th rough the space of dynamica l systems . As corollary results, we indicate how to est imate the mi nimum embedding dimension, extrinsic noise level, metric entropy, an d Lyapunov spectrum. Nu merical an d experimental app lications demonst rate the method's feasibility and limitations. Extensio ns to est imating pa rametrized fami lies of dynamical systems from bifurcation data and to spatial pattern evolut ion are pre sented. Applications to predicting chaotic dat a and t he desig n of forecasting, learning, and control systems are