Chaotic dynamics of Tai Chi public attention revealed by an integrated framework of horizontal visibility graphs, autoencoders, and sparse identification

Understanding the dynamical patterns of public attention towards traditional cultural practices such as Tai Chi is crucial for cultural heritage preservation and promotion strategies. This study reveals the chaotic dynamics underlying Tai Chi public attention through an integrated analytical framework combining Horizontal Visibility Graphs (HVG), Autoencoder neural networks, and Sparse Identification of Nonlinear Dynamics (SINDy). Using daily Baidu Index data (2014–2024) from four representative Chinese provinces (Beijing, Shanghai, Guangdong, Henan), we construct HVGs and quantify chaos through degree distribution exponents (λ). Results reveal significant regional heterogeneity: HVG λ ranges from 0.291 (Beijing PC, strong chaos) to 0.437 (Shanghai PC, quasi-periodic), with corresponding maximum Lyapunov exponents of 0.025–0.036 bits/day (Rosenstein method) and 0.016–0.217 bits/day (Wolf method); the consistent positivity across both estimation methods provides cross-validated confirmation of chaotic dynamics. Autoencoder-based dimensionality reduction achieves reconstruction correlations of 0.896–0.918, enabling discrete SINDy to identify sparse governing equations (37–38 active terms from 48 candidates) with normalized root-mean-square errors of 11.7–12.8%. The dynamical conclusions are triangulated across three analytically independent characterizations—HVG topological classification, Lyapunov-based phase space analysis, and SINDy equation structure—which converge on consistent regional rankings, with Beijing exhibiting the strongest chaotic signatures and Henan the most structured quasi-periodic behavior. Phase space analysis confirms diverse attractor geometries: Beijing exhibits space-filling chaotic trajectories while Shanghai displays quasi-periodic structures. Comparative evaluation against ARIMA, VAR, and LSTM baselines within the same latent space confirms that SINDy achieves superior predictive fidelity (NRMSE 17.6–22.4%) while uniquely providing interpretable governing equations and chaos diagnostics inaccessible to alternative approaches. This interdisciplinary framework bridges complex network theory, deep learning, and dynamical systems analysis, offering a rigorous quantitative paradigm for studying the temporal evolution of public attention in cultural phenomena.