Testing dynamical system variables for reconstruction.

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 [L. A. Aguirre, IEEE Trans. Educ. 38, 33 (1995); C. Letellier, L. A. Aguirre, and J. Maquet, Phys. Rev. E 71, 066213 (2005); L. A. Aguirre, S. B. Bastos, M. A. Alves, and C. Letellier, Chaos 18, 013123 (2008); L. A. Aguirre and C. Letellier, Phys. Rev. E 83, 066209 (2011); and E. Bianco-Martinez, M. S. Baptista, and C. Letellier, Phys. Rev. E 91, 062912 (2015)]; 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 are 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 an additional test on how well different signals can predict the ability to reconstruct a dynamical system, I use the fitting error from training a reservoir computer.