Empirical Discovery of Multi-Scale Transfer of Information in Dynamical Systems

TL;DR

The paper addresses how information flows across scales in weak turbulence, aiming to quantify time scales and directional transfer using information-theoretic metrics. It develops the sorted IDTxl (sIDTxl) algorithm, a non-uniform-embedding, greedy approach that maximizes conditional mutual information across lagged time series to infer multiscale couplings, and validates it on coupled Lorenz–Rössler systems before applying it to a Majda–McLaughlin–Tabak weakly turbulent model. The findings reveal asymmetries in information transfer across wavenumbers and time scales, with forward energy transfer typically carrying more information, and show how the turnover-time setting shapes the observed cascade landscape. The work provides a fully nonlinear, data-driven statistical framework for tracking multiscale energy exchange in chaotic and turbulent systems, with potential broad applicability to real-world data and higher-dimensional turbulence.

Abstract

In this work, we quantify the time scales and information flow associated with multiscale energy transfer in a weakly turbulent system. This is done through a greedy optimization algorithm which finds the maximum conditional-mutual information across lagged embeddings of time series localized by wavenumber. For our chosen weakly turbulent system, the algorithm finds asymmetries in the information flow across wavenumbers, reflecting what are typically described as forward and inverse cascades. However, our approach goes beyond typical heuristic arguments and provides quantitative insight into the intricate multi-wave mixing dynamics necessary to maintain the steady statistical state characterizing weak turbulence. Our work then provides a novel, detailed, and fully nonlinear statistical analysis of a weakly turbulent system. The flexibility of our approach points to broader applicability in real-world data coming from chaotic or turbulent dynamical systems.

Figures (11)

• Figure 1: Plot of Lorenz--Rössler system. Top Figures: Lorenz dynamics for $C=1$ (a) and $C=10$ (b). Bottom Figure: Rössler dynamics (c).
• Figure 2: Information across lags $\ell$, $I\left(x_{n, \ell+j}, x_{n}\right)$, for the Lorenz--Rössler system for $C=1$ (a) and $C=10$ (b) with the maximum lag corresponding to the reciprocal of the average maximum Lyapunov exponent $\lambda_{M}$.
• Figure 3: The average values over $N_{tr}=100$ trials of (a) $I^{t,i}$, (b) $I^{t,m}$, (c) $I^{t,f}$, and (d) the maximum lags for the Lorenz--Rössler system for $C=1$. The standard deviations around the averages of $I^{t,m}$ and $I^{t,f}$ are seen in (e) and (f). Figures (a)-(c), (e), and (f) are plotted on a log-scale. The hypothesis-testing threshold is $\alpha_{h}=.05$, the maximum allowed lag is $L_{M}=3$, and the number of nearest neighbors is $k_{n}=3$. Note, coupling from target-to-source is read left-to-right.
• Figure 4: The average values over $N_{tr}=100$ trials of (a) $I^{t,i}$, (b) $I^{t,m}$, (c) $I^{t,f}$, and (d) the maximum lags for the Lorenz--Rössler system for $C=10$. The standard deviations around the averages of $I^{t,m}$ and $I^{t,f}$ are seen in (e) and (f). Figures (a)-(c), (e), and (f) are plotted on a log-scale. The hypothesis-testing threshold is $\alpha_{h}=.05$, the maximum allowed lag is $L_{M}=3$, and the number of nearest neighbors is $k_{n}=3$. Note, coupling from target-to-source is read left-to-right.
• Figure 5: Plot of $|\psi(x,t)|$ for $\frac{2k^{i}_{+}}{\epsilon^{2}} < t < \frac{2k^{i}_{+}}{\epsilon^{2}} + 160$.