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.
Alzahrani et al. (Sat,) studied this question.