Problem

The study of chaotic dynamics in natural and engineered systems traditionally relies on computationally expensive methods for quantifying unpredictability and designing control strategies. Characterizing basins of attraction, which show regions of phase space leading to different asymptotic states, often involves intensive numerical techniques to estimate properties like fractal dimensions or the Wada property. Similarly, controlling transient chaos, particularly through methods like partial control and the computation of safety functions, demands recursive, high-resolution sweeps of the state space. These classical approaches become prohibitive when dealing with large parameter spaces, high-dimensional systems, noisy environments, or scenarios requiring real-time analysis, limiting their practical applicability.

Method

This paper explores how data-driven machine learning approaches can address the computational bottlenecks of classical chaos analysis and control. For basin characterization, convolutional neural networks (CNNs) are employed. These networks are trained on images of basins to learn the nonlinear mapping from the basin geometry to metrics such as fractal dimension, basin entropy, and boundary basin entropy. For the control of transient chaos, specifically the computation of safety functions and safe sets, transformer-based models are utilized. These models are adapted to approximate safety functions directly from short trajectory data, circumventing the need for the recursive evaluations that make traditional methods computationally intensive.

Results

In basin characterization, CNN surrogates demonstrated accuracy comparable to classical algorithms while significantly reducing computational cost. For instance, CNNs achieved errors of approximately 10^-2 for fractal dimension estimation, reducing computation time from several seconds to under one second per basin, an order of magnitude improvement. Regarding transient chaos control, transformer-based models provided accurate safety functions with mean squared errors of about 10^-4 in one-dimensional maps. Critically, these models predict safety functions directly from trajectory segments, avoiding the computationally prohibitive nested recursions of classical algorithms, especially for higher-dimensional systems where they are expected to scale more favorably.

Implications

The integration of machine learning with nonlinear dynamics presents significant opportunities for more scalable and robust interventions in chaotic systems. However, several challenges remain, including the automated identification of complex phase space structures like riddled basins, scaling partial control methods to higher dimensions, and developing robust uncertainty quantification and interpretability for ML surrogates. Future directions involve creating standardized benchmarks and hybrid validation pipelines. The vision is for ML to complement and expand classical chaos analysis and control by enabling faster, more scalable, and interpretable solutions. This would involve embedding physical constraints into ML architectures, using symbolic regression for equation discovery, and employing uncertainty-aware models and visualization techniques to enhance trust and scientific insight. Ultimately, hybrid approaches combining ML speed with classical rigor will be crucial for advancing chaos research and its applications across various scientific and engineering domains.