Measuring chaos in the Lorenz and R\"ossler models: Fidelity tests for reservoir computing

TL;DR

This paper uses symbolic dynamics to quantify chaos in Lorenz and R"ossler models and introduces a fidelity test for reservoir computing to simulate their chaotic properties effectively.

Contribution

It demonstrates that symbolic measures accurately characterize chaos and introduces a fidelity test for reservoir computing to replicate chaotic dynamics.

Findings

Symbolic measures effectively detect stability windows.

Fidelity test validates reservoir computing models.

Complexity measures align with return map analyses.

Abstract

This study is focused on the qualitative and quantitative characterization of chaotic systems with the use of symbolic description. We consider two famous systems: Lorenz and R\"ossler models with their iconic attractors, and demonstrate that with adequately chosen symbolic partition three measures of complexity, such as the Shannon source entropy, the Lempel-Ziv complexity and the Markov transition matrix, work remarkably well for characterizing the degree of chaoticity, and precise detecting stability windows in the parameter space. The second message of this study is to showcase the utility of symbolic dynamics with the introduction of a fidelity test for reservoir computing for simulating the properties of the chaos in both models' replicas. The results of these measures are validated by the comparison approach based on one-dimensional return maps and the complexity measures.