Measuring chaos in the Lorenz and Rössler models: Fidelity tests for reservoir computing

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.