# Stability Analysis of Reservoir Computers Dynamics via Lyapunov Functions

## TL;DR

This paper applies Lyapunov functions to analyze the nonlinear stability of reservoir computers, identifying regions of stability and linking stability to the polynomial structure of the reservoir dynamics.

## Contribution

It introduces a Lyapunov-based method for stability analysis of reservoir computers and analytically determines stability regions for both continuous and discrete systems.

## Key findings

- Training error is lower within the stability region.
- Stability is influenced by the polynomial structure of the reservoir dynamics.
- Nonzero coefficients for odd and even powers are important for polynomial dynamics.

## Abstract

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).