Testing Dynamical System Variables for 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.