Chaos–Fractal–Entropy Dynamics and Regime Switching in Energy and Financial Markets: MS-VECM and MS-VARDL Methods

Understanding complex systems requires analytical tools capable of covering nonlinear dynamics, structural complexity, and informational uncertainty simultaneously. In this context, chaos theory, fractal analysis, and entropy measures provide complementary perspectives for examining any irregular behavior in natural and socio-economic systems. This paper examined the relation between the Geopolitical Risk Index and the World Uncertainty Index to the volatility of West Texas Intermediate crude oil, gold, and Bitcoin over the period October 2010–February 2026. The analysis was motivated by the recent intensification of geopolitical tensions, particularly conflicts involving Iran, the United States, and Israel, which have significantly heightened uncertainty in global energy and financial markets. The empirical analysis first investigated the underlying complexity of the variables using entropy, chaos, and fractionality measures. Results from the Shannon, R-T entropy, Kolmogorov–Sinai complexity, Hurst, H-M and Lo’s R/S statistics, Phillips, and GPH fractionality tests consistently indicate entropy, fractal persistence, and long-range dependence across the series. In addition, the largest Lyapunov exponents and Hurst coefficients confirmed the presence of chaotic dynamics. The results reveal strong regime heterogeneity with geopolitical shocks exerting significantly stronger effects during high-uncertainty periods. Forecast comparisons show that regime-switching models outperform linear specifications, highlighting the importance of fractal and nonlinear dynamics in understanding financial market responses to geopolitical risk.