TAD-Div: A data-driven framework for quantifying the similarity of nonlinear dynamical systems

Quantifying the similarity between nonlinear dynamical systems directly from data is a fundamental challenge. Existing methods often lack robustness to noise and dimensionality or struggle to model the underlying evolutionary rules. To address this, we propose TAD-Div (TCN-Attention-based Dynamics Divergence), a novel framework based on the principle of cross-reconstructability. Central to TAD-Div is the Dynamic Attention (DynAttn) mechanism. By integrating a Temporal Convolutional Network (TCN) to encode local trajectory histories, DynAttn transforms self-attention: Instead of relying on static geometric matching, it matches the local evolutionary dynamics of the systems. This dynamical divergence is then quantified via a symmetric log-ratio of cross- and self-reconstruction errors. Through comprehensive validation on eight classic nonlinear systems and a real-world bearing fault dataset, we demonstrate that TAD-Div achieves a highly consistent balance of performance and robustness. It outperforms diverse baselines in challenging high-dimensional and noisy scenarios, while remaining highly competitive in idealized cases. Our results highlight its acute sensitivity to parameter variations, pronounced noise robustness, and practical applicability in anomaly detection. Ultimately, TAD-Div provides a rigorous tool for comparing complex systems and characterizing their underlying dynamical rules.