Finite-time Lyaponov analysis of a trained reservoir computer

TL;DR

This paper demonstrates that finite-time Lyapunov exponent distributions effectively reveal transition mechanisms in high-dimensional reservoir computers trained on low-dimensional chaotic systems, surpassing traditional analysis methods.

Contribution

The study introduces FTLE distribution analysis as a novel, systematic approach for understanding transition mechanisms in trained reservoir computers.

Findings

FTLE distributions accurately reproduce different chaotic regimes.

Conventional analysis methods provide limited mechanistic insight.

FTLE analysis is effective for identifying interior crises in reservoir dynamics.

Abstract

We use finite-time Lyapunov exponent (FTLE) distributions to probe transition mechanisms in high-dimensional reservoir maps trained on low-dimensional chaotic dynamics across multiple regimes. While trained reservoirs accurately predict critical transitions and regime shifts, conventional analyses based on time series or bifurcation structure provide limited mechanistic insight, since distinct pathways in high dimensions can yield similar outputs. We show that FTLE statistics overcome this limitation. This is particularly important for interior crises, where direct identification of unstable periodic orbit collisions in the reservoir space is infeasible. Using the logistic map as a canonical example exhibiting intermittency, fully developed chaos, and crisis-induced transitions, we demonstrate that although such distinct regimes are difficult to characterize within the high dimensional reservoir space, their FTLE distributions are faithfully reproduced. This establishes FTLE analysis as a systematic and reliable framework for uncovering transition mechanisms in learned reservoir dynamics.