Statistical and Dynamical Analysis of Hidden Attractors in the Fractional Glukhovsky–Dolzhansky System

This study investigates the reliable numerical analysis of chaotic dynamics in the Glukhovsky–Dolzhansky system, which models convective fluid motion in a rotating ellipsoidal cavity. Hidden and self-excited attractors are localized using the numerical continuation method (NCM), Pyragas time-delayed feedback control, and Leonov’s analytical dimension formula following global stability loss. A critical assessment of Lyapunov exponents and Lyapunov dimensions in a finite-time setting shows that positive values over long but finite intervals may incorrectly indicate sustained chaos due to transient effects and shadowing breakdown. Furthermore, we demonstrate that the fractional order γ plays a bidirectional control role: it induces chaotic behavior at ρ=5 for γ<0.94 and suppresses chaos at ρ=15 for γ<0.93. The multifractal spectrum and correlation dimension are used to quantify attractor complexity, where transient chaos exhibits a broader spectrum (Δα≈0.67) compared to sustained chaos (Δα≈0.48). Monte Carlo simulations, Sobol sensitivity analysis, Kaplan–Meier survival analysis, and bootstrap-based hypothesis testing confirm the robustness of the results. Overall, the findings provide a unified framework for analyzing hidden attractors, transient chaos, and fractional-order effects in nonlinear fluid dynamical systems.