Correlation between the Hurst exponent and the maximal Lyapunov exponent: examining some low-dimensional conservative maps
This work addresses the problem of characterizing chaotic dynamics in physics and applied mathematics, but it is incremental as it applies existing methods to specific maps.
The study investigated the correlation between the Hurst exponent and maximal Lyapunov exponent in low-dimensional conservative maps, finding a tight correlation with Spearman rank ρ=0.83 and ρ=0.75 for two maps, and used a machine learning procedure to reproduce the Hurst exponent distribution from the Lyapunov exponent distribution with high detail.
The Chirikov standard map and the 2D Froeschlé 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 $ρ=0.83$ and $ρ=0.75$ for the Chirikov and 2D Froeschlé 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.