A Unified Framework Linking Entropy, Fractal Dimension, and Lyapunov Exponents in Chaotic Dynamics

This study presents a universal operator framework predicting critical transitions in nonlinear systems through the intrinsic nexus of entropy, fractal geometry, and chaos. We derive a unified model (Equation 4) that integrates fractal dimension (Dᵓ), Lyapunov exponents (λᵢ), and entropy (S) into a single predictive equation, justified through connections to the Kolmogorov–Sinai entropy theorem and the Kaplan–Yorke dimension conjecture. Validated via simulation of the logistic map across the full bifurcation cascade and cross‐validated against the Hénon map, the model captures exponential entropy decay at bifurcation points and geometric‐dynamic coupling across chaotic regimes. Statistical analysis confirms a highly significant linear relationship between Shannon entropy and correlation dimension (Pearson r = 0.971, p < 10 −124 ; Spearman ρ = 0.972, p < 10 −124 ). The framework identifies critical transitions with 89% accuracy in logistic map simulations, establishing entropy‐fractal correlations as fundamental early‐warning signals for tipping points in complex systems.