THERMAL SCIENCE

International Scientific Journal

Muflih Alhazmi

,

Fathi M. Dawalbait

,

Abdulrahman F. Aljohani

,

Khdija O. Taha

,

Haroon D.S. Adam

,

Sayed Saber

NUMERICAL APPROXIMATION METHOD AND CHAOS FOR A CHAOTIC SYSTEM IN SENSE OF CAPUTO-FABRIZIO OPERATOR

ABSTRACT
This paper presents a novel numerical method for analvwing chaotic systems, focusing on applications to real-world problems. The Caputo-Fabrizio operator, a fractional derivative without a singular kernel, is used to investigate chaotic behavior. A fractional-order chaotic model is analvwed using numerical solutions derived from this operator, which captures the complexity of chaotic dynamics. In this paper, the uniqueness and boundedness of the solution are established using fixed-point theory. Due to the non-linearity of the system, an appropriate numerical scheme is developed. We further explore the model's dynamical properties through phase portraits, Lyapunov exponents, and bifurcation diagrams. These tools allow us to observe the system’s sensitivity to varying parameters and derivative orders. Ultimately, this work extends the application of fractional calculus to chaotic systems and provides a robust methodology for obtaining insights into complex behaviors.
KEYWORDS
fractional derivatives, non-linear equations, simulation, numerical results, iterative method, time varying control system, Lyapunov functions
PAPER SUBMITTED: 2024-06-12
PAPER REVISED: 2024-09-10
PAPER ACCEPTED: 2024-10-11
PUBLISHED ONLINE: 2025-01-25
DOI REFERENCE: https://doi.org/10.2298/TSCI2406161A
CITATION EXPORT: view in browser or download as text file
THERMAL SCIENCE YEAR 2024, VOLUME 28, ISSUE Issue 6, PAGES [5161 - 5168]
REFERENCES
  1. Atangana, A., Bonyah, E., Fractional Stochasticmodelling: New Approach to Capture More Hetero-Geneity, Chaos, 29 (2019), 1, 013118
  2. Caputo, M., Linear Model of Dissipation Whose Q Is Almost Frequency Independent-II, Geophys J. Int., 13 (1967), 5, pp. 529-539
  3. Atangana, A., Araz, I. S., New Numerical Method for Ordinary Differential Equations: Newton Polyno-Mial, J. Comput. Appl. Math., 372 (2019), 112622
  4. Basios, V., Antonopoulos, C. G., Hyperchaos and Labyrinth Chaos: Revisiting Thomas-Rossler Systems, J. Theor. Biol., 460 (2019), Jan., pp. 153-159
  5. Le Berre, M., et al., Example of a Chaotic Crystal: The Labyrinth, Phys. Rev. E66, (2002), 026203
  6. Xin, B. G., et al., A Fractional Model of Labyrinth Chaos and Numeri-Cal Analysis, Int. J. Non-linear Sci. Numer. Simul., 11 (2010), 10, pp. 837-842
  7. Sprott, J. C., Chlouverakis, K. E., Labyrinth Chaos, Int. J. Bifur. Chaos, 17 (2007), 6, pp. 2097-2108
  8. Thomas, R., Deterministic Chaos Seen in Termsof Feedback Circuits: Analysis, Synthesis, Labyrinthchaos, Int. J. Bifur. Chaos, 9 (1999), 10, pp. 1889-1905
  9. Caputo, M., Fabrizio, M., A New Definition of Fractional Derivative Without Singular Kernel, Prog. Fract. Differ. Appl., 1 (2015), 2, pp. 73-85
  10. Caputo, M., Fabrizio, M., On the Notion of Fractional Derivative and Applications to The Hysteresis Phenomena, Mecc, 52 (2017), 13, 3043-3052
  11. Almutairi, N., Saber, S., On Chaos Control of Non-Linear Fractional Newton-Leipnik System Via Fractional Caputo-Fabrizio Derivatives, Sci. Rep., 13 (2023), 22726
  12. Salem M. A., et al., Modelling COVID-19 Spread and Non-Pharmaceutical Interventions in South Africa: A Stochastic Approach, Scientific African, 24 (2024), e02155
  13. 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
  14. 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
  15. 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
  16. Almutairi, N., Saber, S., Existence of Chaos and the Approximate Solution of the Lorenz-Lü-Chen System with the Caputo Fractional Operator, AIP Advances, 14 (2024), 1, 015112
  17. Almutairi, N., Sayed, 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
  18. 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
  19. Danca, M. F., Lyapunov Exponents of a Discontinuous 4-D Hyperchaotic System of Integer or Fractional Order, Entr., 20 (2018), 5, 337
  20. Danca, M. F., Kuznetsov, N., Matlab Code for Lyapunov Exponents of Fractional-Order Systems, Int. J. Bif. Chaos, 28 (2018), 5, 1850067
PDF VERSION [DOWNLOAD]
Numerical approximation method and chaos for a chaotic system in sense of Caputo-Fabrizio operator

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