Generalized Controllability, Stability, and Chaos in Fractional ‎Dynamics: A Unified Approach

  • Authors

    • Thwiba A. Khalid Department of Mathematics, Faculty of Science, Al-Baha University, Al-Baha, 65779, Saudi Arabia and College of Engineering and Medical Sciences, Khartoum 12045, Sudan
    https://doi.org/10.14419/yfbc5s43

    Received date: December 8, 2025

    Accepted date: December 26, 2025

    Published date: January 2, 2026

  • Fractional Calculus; Chaos Control; Adaptive Algorithms; Nonlinear Dynamic; Stability Analysis; Predictor-Corrector Methods; Lyapunov ‎Exponents.

  • Abstract

    This paper introduces a unified analytical and computational framework for controlling and characterizing chaos in generalized fractional-‎order dynamical systems. Despite advances in fractional calculus, existing methods face persistent challenges: high computational costs, ‎limited stability criteria, and a lack of quantitative chaos measures for generalized operators. To bridge these gaps, we propose Apc-GM, a ‎novel adaptive predictor-corrector algorithm that enhances numerical accuracy by 45% and reduces computation time by 23% compared to ‎classical approaches. Our framework integrates three core pillars: a generalized controllability theory, an extended Lyapunov stability ‎analysis valid for all orders α>0, and a quantitative chaos criterion based on generalized Lyapunov exponents. Numerical validation on a 4D ‎fractional Lorenz-Stenflo system demonstrates 92% chaos suppression efficiency under optimal control, while maintaining consistency with ‎classical results. This work provides robust, ready-to-use tools for modeling, analysis, and control of complex fractional-order systems in ‎engineering and applied sciences. Keywords: fractional calculus, chaos control, adaptive numerical methods, nonlinear dynamics, stability ‎analysis, Lyapunov exponents, predictor-corrector methods.

  • References

    1. Aguila-Camacho, N., Duarte-Mermoud, M. A., & Gallegos, J. A. (2014). Lyapunov functions for fractional order systems. Communications in Nonlinear Science and Numerical Simulation, 19(9), 2951-2957. https://doi.org/10.1016/j.cnsns.2014.01.022.
    2. Atangana, A., & Baleanu, D. (2016). New fractional derivatives with non-local and non-singular kernel: Theory and application to heat transfer model. Thermal Science, 20(2), 763-769. https://doi.org/10.2298/TSCI160111018A.
    3. Baleanu, D., & Wu, G. C. (2023). Fractional calculus in viscoelasticity: Recent advances and applications. Applied Mathematical Modelling, 114, 502-516.
    4. Caputo, M., & Fabrizio, M. (2015). A new definition of fractional derivative without singular kernel. Progress in Fractional Differentiation and Applications, 1(2), 73-85.
    5. Shafique, A., Kolev, G., Bayazitov, O., Bobrova, Y., & Kopets, E. (2025). Chaos in Control Systems: A Review of Suppression and Induction Strategies with Industrial Applications. Mathematics, 13(24), 4015.‏ https://doi.org/10.3390/math13244015.
    6. Diethelm, K., Ford, N. J., & Freed, A. D. (2002). A predictor-corrector approach for the numerical solution of fractional differential equations. Nonlinear Dynamics, 29(1-4), 3-22. https://doi.org/10.1023/A:1016592219341.
    7. Alsaedy, A. F., Alshabhi, S. H., Mohammed, M. M., Khalid, T. A., Mustafa, A. O., Bashir, R. A., ... & Bakery, A. A. (2024). Decision-making on A Novel Stochastic Space-I of Solutions for Fuzzy Volterra-Type Non-linear Dynamical Economic Models. Contemporary Mathematics, 6320-6340.‏ https://doi.org/10.37256/cm.5420245499.
    8. Abdul-Kareem, A. A., & Al-Jawher, W. A. M. (2023). A Hybrid Domain Medical Image Encryption Scheme Using URUK and WAM Chaotic Maps with Wavelet-Fourier Transforms. Journal of Cyber Security and Mobility, 12(4), 435-464.‏ https://doi.org/10.13052/jcsm2245-1439.1241.
    9. Nasrolahpour, H., Pellegrini, M., & Skovranek, T. (2025). Fractional Calculus in Epigenetics: Modelling DNA Methylation Dynamics Using Mittag–Leffler Function. Fractal and Fractional, 9(9), 616.‏ https://doi.org/10.3390/fractalfract9090616.
    10. Alfihed, S., Majrashi, M., Ansary, M., Alshamrani, N., Albrahim, S. H., Alsolami, A., ... & Al-Otaibi, F. (2024). Non-invasive brain sensing technologies for modulation of neurological disorders. Biosensors, 14(7), 335.‏ https://doi.org/10.3390/bios14070335.
    11. Alshabhi, S. H., Mohamed, O. K. S., Mohammed, M. M., Khalid, T. A., Mustafa, A. O., Magzoub, M., . . . & Bakery, A. A. (2024). Decision-Making of Fredholm Operator on a New Variable Exponents Sequence Space of Supply Fuzzy Functions Defined by Leonardo Numbers. Contemporary Mathematics, 5534-5545. https://doi.org/10.37256/cm.5420245619.
    12. Petráš, I. (2011). Fractional-order nonlinear systems: modeling, analysis and simulation. Springer Science & Business Media. https://doi.org/10.1007/978-3-642-18101-6.
    13. Learn, R., & Feigenbaum, E. (2016). Adaptive step-size algorithm for Fourier beampropagation method with absorbing boundary layer of auto-determined width. Ap plied Optics, 55(16), 4402-4407. https://doi.org/10.1364/AO.55.004402.
    14. Klamka, J., Babiarz, A., Czornik, A., & Niezabitowski, M. (2020). Controllability and stability of semilinear fractional order systems. In Automatic Control, Robotics, and Information Processing (pp. 267-290). Cham: Springer International Publishing. https://doi.org/10.1007/978-3-030-48587-0_9.
    15. ‏Abed-Elhameed, T. M., & Aboelenen, T. (2022). Mittag-Leffler stability, control, and synchronization for chaotic generalized fractional-order systems. Advances in Continuous and Discrete Models, 2022:50. https://doi.org/10.1186/s13662-022-03721-9.
    16. Odibat, Z., & Baleanu, D. (2020). Numerical simulation of initial value problems with generalized Caputo-type fractional derivatives. Applied Numerical Mathematics, 156, 94-105. https://doi.org/10.1016/j.apnum.2020.04.015.
    17. Khalid, T. Application of Elzaki Transform Decomposition Method in Solving TimeFractional Sawada Kotera Ito Equation. Malaysian Journal of Mathematical Sciences. 19 (2025) https://doi.org/10.47836/mjms.19.2.17.
    18. Ogbumba, R. O., Shagari, M. S., Alansari, M., Khalid, T. A., Mohamed, E. A., & Bakery, A. A. (2023). Advancements in hybrid fixed point results and F-contractive operators. Symmetry, 15(6), 1253. https://doi.org/10.3390/sym15061253.
    19. Khalid, T., Alnoor, F., Babeker, E., Ahmed, E. & Mustafa, A. Legendre polynomials and techniques for collocation in the computation of variable-order fractional advection-dispersion equations. International Journal of Analysis and Applications. 22 pp. 185-185 (2024) https://doi.org/10.28924/2291-8639-22-2024-185.
    20. Hamadneh, T., Hioual, A., Saadeh, R., Abdoon, M. A., Almutairi, D. K., Khalid, T. A., & Ouannas, A. (2023). General methods to synchronize fractional discrete reaction-diffusion systems applied to the glycolysis model. Fractal and Fractional, 7(11), 828. . https://doi.org/10.3390/fractalfract7110828.
    21. Nasrolahpour, H., Pellegrini, M., & Skovranek, T. (2025). Fractional Calculus in Epigenetics: Modelling DNA Methylation Dynamics Using Mittag–Leffler Function. Fractal and Fractional, 9(9), 616. https://doi.org/10.3390/fractalfract9090616.
    22. Sharma, S., Raj, K., Sharma, S. K., Khalid, T. A., Mustafa, A. O., Mohammed, M. M., ... & Bakery, A. A. (2024). Applications of strongly deferred weighted convergence in the environment of uncertainty. International Journal of Analysis and Applications, 22, 181-181.‏ https://doi.org/10.28924/2291-8639-22-2024-181.
    23. Alsolmi, M. M., Alshabhi, S. H., Mohammed, M., Khalid, T. A., Magzoub, M., Taha, N. E., ... & Bakery, A. A. (2025). On a New Stochastic Space with Applications to Nonlinear Economic Models. European Journal of Pure and Applied Mathematics, 18(1), 5641-5641.‏ https://doi.org/10.29020/nybg.ejpam.v18i1.5641.
    24. Gao, X., Li, Y., Liu, X., Ye, Y., & Fan, H. (2023). Stability analysis of fractional bidirectional associative memory neural networks with multiple proportional delays and distributed delays. IEEE Transactions on Neural Networks and Learning Systems.
    25. Baleanu, D., Sajjadi, S. S., Jajarmi, A., & Defterli, Ö. (2021). On a nonlinear dynamical system with both chaotic and nonchaotic behaviors: a new fractional analysis and control. Advances in Difference Equations, 2021(1), 234.‏ https://doi.org/10.1186/s13662-021-03393-x.
    26. Meghdari, A. (2014). Identification of 4D Lü hyper-chaotic system using identical systems synchronization and fractional adaptation law. Applied Mathematical Modelling.
    27. Petras, I. (2021). Fractional-Order Nonlinear Systems: Modeling, Analysis and Simulation. Springer.
    28. Zhang, Q. H., Lu, J. G., & Zhu, Z. (2025). A review on robust control of continuous time fractional-order systems: QH Zhang et al. Nonlinear Dynamics, 1-24.
    29. Baleanu, D., & Wu, G. C. (2023). Fractional calculus in viscoelasticity: Recent advances and applications. Applied Mathematical Modelling, 114, 502-516.
    30. Vieira, L. C., Costa, R. S., & Valério, D. (2023). An overview of mathematical modelling in cancer research: fractional calculus as modelling tool. Fractal and fractional, 7(8), 595.‏ https://doi.org/10.3390/fractalfract7080595.
    31. Abdulrhman, T. Stability Analysis of Fractional Chaotic and Fractional-Order Hyperchain Systems Using Lyapunov Functions. European Journal of Pure and Applied Mathematics. 18, 5576-5576 (2025) https://doi.org/10.29020/nybg.ejpam.v18i1.5576.
    32. Caputo, M., & Fabrizio, M. (2015). A new definition of fractional derivative without singular kernel. Progress in Fractional Differentiation and Applications, 1(2), 73-85.
    33. Atangana, A., & Baleanu, D. (2016). New fractional derivatives with non-local and non-singular kernel: Theory and application to heat transfer model. Thermal Science, 20(2), 763-769. https://doi.org/10.2298/TSCI160111018A.
    34. Kilbas, A. A., Srivastava, H. M., & Trujillo, J. J. (2006). Theory and Applications of Fractional Differential Equations. Elsevier.
    35. Khalid, T. Dynamics of Predator-Prey Interactions, Analyzing the Effects of Time Delays and Neymark-Saker Bifurcation. International Journal of Neutrosophic Science (IJNS). 26 (2025) https://doi.org/10.54216/IJNS.260224.
    36. Tenreiro Machado, J. A. (2003). A probabilistic interpretation of the fractional-order differentiation.‏
    37. Samko, S. G., Kilbas, A. A., & Marichev, O. I. (1993). Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach.
    38. Mainardi, F. (2010). An introduction to mathematical models. Fractional calculus and waves in linear viscoelasticity. Imperial College Press, London‏. https://doi.org/10.1142/p614.

  • Downloads

  • How to Cite

    Khalid, T. A. . (2026). Generalized Controllability, Stability, and Chaos in Fractional ‎Dynamics: A Unified Approach. International Journal of Basic and Applied Sciences, 14(8), 652-659. https://doi.org/10.14419/yfbc5s43