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Fractal Fract
Volume 10
Issue 6
10.3390/fractalfract10060377
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This is an early access version, the complete PDF, HTML, and XML versions will be available soon.
Article

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

by
Salem Mubarak Alzahrani
1,
Ghaliah Alhamzi
2,
Mona Bin-Asfour
2,*,
Mansoor Alsulami
3,
Khdija O. Taha
4,
Najat Almutairi
4 and
Sayed Saber
1,5
1
Department of Mathematics, Faculty of Science, Al-Baha University, Al-Baha 65779, Saudi Arabia
2
Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh 11623, Saudi Arabia
3
Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia
4
Department of Mathematics, College of Science, Qassim University, Buraidah 51452, Saudi Arabia
5
Department of Mathematics and Computer Science, Faculty of Science, Beni-Suef University, Beni-Suef 62511, Egypt
*
Author to whom correspondence should be addressed.
Fractal Fract. 2026, 10(6), 377; https://doi.org/10.3390/fractalfract10060377
Submission received: 29 March 2026 / Revised: 1 May 2026 / Accepted: 12 May 2026 / Published: 30 May 2026
(This article belongs to the Special Issue Advances in Fractal and Fractional Dynamics)
Versions Notes

Abstract

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.
Keywords: global stability; chaos; hidden attractor; transient set; Lyapunov exponents; Lyapunov dimension; unstable periodic orbit; time-delayed feedback control; convective fluid motion global stability; chaos; hidden attractor; transient set; Lyapunov exponents; Lyapunov dimension; unstable periodic orbit; time-delayed feedback control; convective fluid motion

Share and Cite

MDPI and ACS Style

Alzahrani, S.M.; Alhamzi, G.; Bin-Asfour, M.; Alsulami, M.; Taha, K.O.; Almutairi, N.; Saber, S. Statistical and Dynamical Analysis of Hidden Attractors in the Fractional Glukhovsky–Dolzhansky System. Fractal Fract. 2026, 10, 377. https://doi.org/10.3390/fractalfract10060377

AMA Style

Alzahrani SM, Alhamzi G, Bin-Asfour M, Alsulami M, Taha KO, Almutairi N, Saber S. Statistical and Dynamical Analysis of Hidden Attractors in the Fractional Glukhovsky–Dolzhansky System. Fractal and Fractional. 2026; 10(6):377. https://doi.org/10.3390/fractalfract10060377

Chicago/Turabian Style

Alzahrani, Salem Mubarak, Ghaliah Alhamzi, Mona Bin-Asfour, Mansoor Alsulami, Khdija O. Taha, Najat Almutairi, and Sayed Saber. 2026. "Statistical and Dynamical Analysis of Hidden Attractors in the Fractional Glukhovsky–Dolzhansky System" Fractal and Fractional 10, no. 6: 377. https://doi.org/10.3390/fractalfract10060377

APA Style

Alzahrani, S. M., Alhamzi, G., Bin-Asfour, M., Alsulami, M., Taha, K. O., Almutairi, N., & Saber, S. (2026). Statistical and Dynamical Analysis of Hidden Attractors in the Fractional Glukhovsky–Dolzhansky System. Fractal and Fractional, 10(6), 377. https://doi.org/10.3390/fractalfract10060377

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