Correlation between the Hurst exponent and the maximal Lyapunov exponent: examining some low-dimensional conservative maps

TL;DR

This study reveals a strong correlation between the Hurst exponent and the maximal Lyapunov exponent in low-dimensional conservative maps, and demonstrates that machine learning can effectively predict one from the other.

Contribution

It uncovers a significant correlation between HE and mLE in conservative maps and introduces a machine learning approach to predict HE from mLE data.

Findings

Strong correlation between HE and mLE (Spearman's ρ > 0.75)

ML successfully reproduces HE distribution from mLE data

Structures in parameter space are captured in detail by ML predictions

Abstract

The Chirikov standard map and the 2D Froeschl\'e map are investigated. A few thousand values of the Hurst exponent (HE) and the maximal Lyapunov exponent (mLE) are plotted in a mixed space of the nonlinear parameter versus the initial condition. Both characteristic exponents reveal remarkably similar structures in this space. A tight correlation between the HEs and mLEs is found, with the Spearman rank $\rho=0.83$ and $\rho=0.75$ for the Chirikov and 2D Froeschl\'e maps, respectively. Based on this relation, a machine learning (ML) procedure, using the nearest neighbor algorithm, is performed to reproduce the HE distribution based on the mLE distribution alone. A few thousand HE and mLE values from the mixed spaces were used for training, and then using $2-2.4\times 10^5$ mLEs, the HEs were retrieved. The ML procedure allowed to reproduce the structure of the mixed spaces in great detail.