Stability analysis of reservoir computers dynamics via Lyapunov functions

A Lyapunov design method is used to analyze the nonlinear stability of a generic reservoir computer for both the cases of continuous-time and discrete-time dynamics. Using this method, for a given nonlinear reservoir computer, a radial region of stability around a fixed point is analytically determined. We see that the training error of the reservoir computer is lower in the region where the analysis predicts global stability but is also affected by the particular choice of the individual dynamics for the reservoir systems. For the case that the dynamics is polynomial, it appears to be important for the polynomial to have nonzero coefficients corresponding to at least one odd power (e.g., linear term) and one even power (e.g., quadratic term).