THERMAL SCIENCE

International Scientific Journal

Tao Yan

,

Muflih Alhazmi

,

Mukhtar Y. Youssif

,

Amna E. Elhag

,

Abdulrahman F. Aljohani

,

Sayed Saber

ANALYSIS OF A LORENZ MODEL USING ADOMIAN DECOMPOSITION AND FRACTAL-FRACTIONAL OPERATORS

ABSTRACT
This paper extends the classical Lorenz system to incorporate fractal-fractional dynamics, providing a detailed numerical analysis of its chaotic behavior. By applying Caputo's fractal-fractional operators to the Lorenz system, the study explores the fractal and fractional nature of non-linear systems. Numerical methods are employed to solve the extended system, with suitable fractal and fractional orders chosen to demonstrate chaos and hyper-chaos. The results are presented graphically, highlighting the complex dynamic behavior of the system under different parameter conditions. This research advances the understanding of fractional calculus in modelling and controlling chaotic systems in various scientific fields.
KEYWORDS
fractional derivatives, non-linear equations, simulation, numerical results, iterative method, time varying control system, Lyapunov functions
PAPER SUBMITTED: 2024-07-19
PAPER REVISED: 2024-10-20
PAPER ACCEPTED: 2024-11-13
PUBLISHED ONLINE: 2025-01-25
DOI REFERENCE: https://doi.org/10.2298/TSCI2406001Y
CITATION EXPORT: view in browser or download as text file
THERMAL SCIENCE YEAR 2024, VOLUME 28, ISSUE Issue 6, PAGES [5001 - 5009]
REFERENCES
  1. Arena, P., et al., Non-Linear Non-Integer Order Circuits and Systems, World Scientific, Singapore, Singapore, 2000
  2. Hilfer, R., Applications of fractional Calculus in Physics, World Scientific, Singapore, Singapore, 2000
  3. Ahmed, E., Elgazzar A. S., On Fractional Order Differential Equations Model for Non-Local Epidemics, Physica A., 379 (2007), 2, pp. 607-614
  4. Ahmad, W. M., El-Khazali R., Fractional-Order Dynamical Models of Love, Chaos Soliton Fract, 33 (2007), 4, pp. 1367-1375
  5. Bagley, R. L., Calico, R. A., Fractional Order State Equations for the Control of Viscoelastically Damped Structures, J. Guid Control Dyn, 14 (1991), May, pp. 304-311
  6. Sun, H. H., et al., Linear Approximation of Transfer Function with a Pole of Fractional Order, IEEE Trans Auto Contr, 29 (1984), 5, pp. 441-444
  7. Ichise, M., et al., An Analog Simulation of Non-Integer Order Transfer Functions for Analysis of Electrode process, J. Electroanal Chem., 33 (1971), 2, pp. 253-265
  8. Heaviside, O., Electromagnetic Theory, Chelsea, New York, USA, 1971
  9. Kusnezov, D., et al., Quantum Levy Processes and Fractional Kinetics, Phys. Re.v Lett., 82 (1999), 1136
  10. Podlubny, I., Geometric and Physical Interpretation of Fractional Integration and Fractional Differentiation, Fract. Calc. Appl. Anal., 5 (2002), 4, pp. 367-386
  11. Grigorenko, I., Grigorenko E., Chaotic Dynamics of the Fractional Lorenz System, Phys. Rev. Lett., 91 (2003), 034101
  12. Almutairi, N., Saber, S., Existence of Chaos and the Approximate Solution of the Lorenz-Lu-Chen System with the Caputo Fractional Operator, AIP Advances, 14 (2024), 1, 015112
  13. Hartley, T. T., et al., Chaos in a Fractional Order Chua's system, IEEE Trans. Circ. Syst. I, 42 (1995), 8, pp. 485-490
  14. Li, C. G., Chen, G., Chaos and Hyperchaos in the Fractional-Order Rossler Equations, Physica A., 341 (2004), Oct., pp. 55-61
  15. Li, C. P., Peng, G. J., Chaos in Chen's System with a Fractional Order, Chaos Soliton. Fract., 22 (2004), 2, pp. 443-450
  16. Wang, X. Y., Wang, M. J., Dynamic Analysis of the Fractional-Order Liu System and Its Synchronization, Chaos, 17 (2007), 033106
  17. Ahmed, K. I. A., et al., Analytical Solutions for a Class of Variable-Order Fractional Liu System under Time-Dependent Variable Coefficients, Results in Physics, 56 (2024), 107311
  18. Saber, S., Control of Chaos in the Burke-Shaw System of Fractal-Fractional Order in the Sense of Caputo-Fabrizio, Journal of Applied Mathematics and Computational Mechanics, 23 (2024), 1, pp. 83-96
  19. Almutairi, N., Saber, S., On Chaos Control of Non-Linear Fractional Newton-Leipnik System Via Fractional Caputo-Fabrizio Derivatives, Sci. Rep., 13 (2023), 22726
  20. Salem M. A., et al., Modelling COVID-19 Spread and Non-Pharmaceutical Interventions in South Africa: A Stochastic Approach, Scientific African, 24 (2024), e02155
  21. Ahmed, K. I. A., et al., Different Strategies for Diabetes by Mathematical Modelling: Applications of Fractal-Fractional Derivatives in the Sense of Atangana-Baleanu, Results Phys., 52 (2023), 106892
  22. Salem M. A., et al., Numerical Simulation of an Influenza Epidemic, Prediction with Fractional SEIR and the ARIMA Model, 18 (2024), 1, pp. 1-12
  23. Rania et al., Mathematical Modelling and Stability Analysis of the Novel Fractional Model in the Caputo Derivative Operator, A Case Study Saadeh, Heliyon, 10 (2024), 5, e26611
  24. Almutairi, N., Saber, S., Chaos Control and Numerical Solution of Time-Varying Fractional Newton-Leipnik System Using Fractional Atangana-Baleanu Derivatives, AIMS Mathematics, 8 (2023), 11, pp. 25863-25887
  25. Caputo, M., Fabrizio, M., A New Definition of Fractional Derivative without Singular Kernel, Prog. Fract. Differ. Appl., 1 (2015), 2, pp. 73-85
  26. Almutairi, N., Saber, S., Application of a Time-Fractal Fractional Derivative with a Power-Law Kernel to the Burke-Shaw System Based on Newton's Interpolation Polynomials, MethodsX, 12 (2024), 102510
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Analysis of a Lorenz model using adomian decomposition and fractal-fractional operators

2025 Society of Thermal Engineers of Serbia. Published by the Vinča Institute of Nuclear Sciences, National Institute of the Republic of Serbia, Belgrade, Serbia. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution-NonCommercial-NoDerivs 4.0 International licence