Applications of Lyapunov Exponents, Correlation Dimension, and Kolmogorov Entropy in Chaotic Dynamical Systems

Lyapunov exponents, correlation dimension, and Kolmogorov entropy are widely used metrics in chaotic dynamical systems. The Lyapunov exponent quantifies system stability by measuring the exponential divergence rate of nearby trajectories in phase space, which can be computationally implemented using Jacobian matrix calculations or Wolf's algorithm for time series data. The correlation dimension characterizes the phase space structure by analyzing the scaling properties of point distributions, typically computed through Grassberger-Procaccia algorithm involving pairwise distance calculations and log-log plots. Kolmogorov entropy describes the degree of chaos by measuring the rate of information loss in the system, often estimated using nearest neighbor methods or partition-based approaches. These metrics play crucial roles in exploring chaotic phenomena, predicting system dynamic behaviors, and designing control strategies. Scholars like Chen have made significant contributions to applying these indicators, establishing new research directions for chaotic dynamical systems through innovative numerical implementations and practical applications.