Testing Dynamical System Variables for Reconstruction

TL;DR

This paper compares nonlinear observability and a continuity statistic to evaluate the suitability of different variables for reconstructing dynamical systems, introducing reservoir computer fitting error as an additional predictive measure.

Contribution

It introduces a comparative analysis of nonlinear observability and continuity statistics for dynamical system reconstruction, incorporating reservoir computer error as a new predictive metric.

Findings

Nonlinear observability and continuity provide ambiguous predictions without a metric.

Reservoir computer fitting error correlates with reconstruction quality.

The methods help identify suitable variables for system reconstruction.

Abstract

Analyzing data from dynamical systems often begins with creating a reconstruction of the trajectory based on one or more variables, but not all variables are suitable for reconstructing the trajectory. The concept of nonlinear observability has been investigated as a way to determine if a dynamical system can be reconstructed from one signal or a combination of signals, however nonlinear observability can be difficult to calculate for a high dimensional system. In this work I compare the results from nonlinear observability to a continuity statistic that indicates the likelihood that there is a continuous function between two sets of multidimensional points- in this case two different reconstructions of the same attractor from different signals simultaneously measured. Without a metric against which to test the ability to reconstruct a system, the predictions of nonlinear observability and continuity are ambiguous. As a additional test how well different signals can predict the ability to reconstruct a dynamical system I use the fitting error from training a reservoir computer.