Skip to main content
SpringerLink
Log in

Effective methods for numerical analysis of the simplest chaotic circuit model with Atangana–Baleanu Caputo fractional derivative

  • Published:
Journal of Engineering Mathematics Aims and scope Submit manuscript

Abstract

This paper comprehensively studies effective numerical methods for solving the simplest chaotic circuit model. We introduce a novel scheme for the Atangana–Baleanu Caputo fractional derivative (ABC-FD), coupled with the Laplace decomposition method (LDM). Furthermore, we rigorously compare the performance of these proposed methods with the Runge–Kutta fourth-order method. Using two mathematical techniques, we have discovered effective and highly convergent solutions to the chaotic model. We gave different values to the parameters to plot the chaos and create a phase portrait of the system. Therefore, the provided methods can be applied to more sophisticated examinations of different models. This study advances numerical techniques for understanding chaotic dynamics in complex systems. By introducing a novel scheme for the Atangana–Baleanu Caputo fractional derivative and the Laplace decomposition method, we provide a robust framework for effectively solving the simplest chaotic circuit model. This framework enhances accuracy and efficiency in unraveling chaotic behaviors, contributing to a broader understanding of chaotic dynamics across scientific domains in the future.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others

Data availability

No data were used to support this study.

References

  1. Podlubny I (1999) Fractional differential equations. Academic Press, New York

    Google Scholar 

  2. Oldham KB, Spanier J (1974) The fractional calculus. Academic Press, New York

    Google Scholar 

  3. Gorenflo R, Mainardi F (1997) Fractional calculus. In: Carpinteri A, Mainardi F (eds) Fractals and fractional calculus in continuum mechanics, vol 378. International Centre for Mechanical Sciences. Springer, Vienna

    Google Scholar 

  4. Samko G, Kilbas A, Marichev O (1993) fractional integrals and derivatives: theory and applications. Gordon and Breach, Amsterdam

    Google Scholar 

  5. Dudkowski D, Jafari S, Kapitaniak T, Kuznetsov NV, Leonov GA, Prasad A (2016) Hidden attractors in dynamical systems. Phys Rep 637:1–50

    Article  ADS  MathSciNet  Google Scholar 

  6. Xu G, Shekofteh Y, Akgül A, Li C, Panahi S (2018) A new chaotic system with a self-excited attractor: entropy measurement, signal encryption, and parameter estimation. Entropy 20(2):86

    Article  ADS  PubMed  PubMed Central  Google Scholar 

  7. Lai Q, Akgul A, Li C, Xu G, Çavuşoğlu Ü (2017) A new chaotic system with multiple attractors: dynamic analysis, circuit realization and S-Box design. Entropy 20(1):12

    Article  ADS  PubMed  PubMed Central  Google Scholar 

  8. Petráš I (2008) A note on the fractional-order Chua’s system. Chaos Solitons Fractals 38(1):140–147

    Article  ADS  Google Scholar 

  9. Abd El-Maksoud AJ, Abd El-Kader AA, Hassan BG, Rihan NG, Tolba MF, Said LA, Radwan AG, Abu-Elyazeed MF (2019) fpga implementation of sound encryption system based on fractional-order chaotic systems. Microelectronics 90:323–35

    Article  Google Scholar 

  10. Kumar S, Kumar R, Cattani C, Samet B (2020) Chaotic behaviour of fractional predator–prey dynamical system. Chaos Solitons Fractals 135(109811):109811

    Article  MathSciNet  Google Scholar 

  11. Toufik M, Atangana A (2017) New numerical approximation of fractional derivative with non-local and non-singular kernel: application to chaotic models. Eur Phys J Plus 132(10):444

    Article  Google Scholar 

  12. Abdoon MA, Saadeh R, Berir M, El Guma F, Ali M (2023) Analysis, modeling and simulation of a fractional-order influenza model. Alexandr Eng J 74:231–40

    Article  Google Scholar 

  13. Li P, Han L, Xu C, Peng X, ur Rahman M, Shi S (2023) Dynamical properties of a meminductor chaotic system with fractal–fractional power law operator. Chaos Solitons Fractals 175:114040

    Article  MathSciNet  Google Scholar 

  14. ur Rahman M, El-Shorbagy MA, Alrabaiah H, Baleanu D, la sen MD (2023) Investigating a new conservative 4-dimensional chaotic system. Results Phys 53:106969

    Article  Google Scholar 

  15. Du S, Haq NU, Rahman MU (2023) Novel multiple solitons, their bifurcations and high order breathers for the novel extended Vakhnenko–Parkes equation. Results Phys 54:107038

    Article  Google Scholar 

  16. ur Rahman M (2022) Generalized fractal–fractional order problems under non-singular Mittag–Leffler kernel. Results Phys 35:105346

    Article  Google Scholar 

  17. Qu H, ur Rahman M, Ahmad S, Riaz MB, Ibrahim M, Saeed T (2022) Investigation of fractional order bacteria dependent disease with the effects of different contact rates. Chaos Solitons Fractals 159:112169

    Article  MathSciNet  Google Scholar 

  18. Alzahrani ABM, Abdoon MA, Elbadri M, Berir M, Elgezouli DE (2023) A Comparative numerical study of the symmetry chaotic jerk system with a hyperbolic sine function via two different methods. Symmetry 15(11):1991

    Article  ADS  Google Scholar 

  19. Ghanbari B, Atangana A (2020) Some new edge detecting techniques based on fractional derivatives with nonlocal and non-singular kernels. Adv Differ Equ 2020(1):1–19

    Article  Google Scholar 

  20. Sene N (2020) Second-grade fluid model with Caputo–Liouville generalized fractional derivative. Chaos Solitons Fractals 133(109631):109631

    Article  MathSciNet  Google Scholar 

  21. Petras I (2002) Control of fractional-order Chua’s system. J Electron Eng 53(8):219–222

    Google Scholar 

  22. Elbadri M (2022) Initial value problems with generalized fractional derivatives and their solutions via generalized Laplace decomposition method. Adv Math Phys 2022:1–7

    Article  MathSciNet  Google Scholar 

  23. Elbadri M, Ahmed SA, Abdalla YT, Hdidi W (2020) A new solution of time-fractional coupled KdV equation by using natural decomposition method. Abstr Appl Anal 2020:1–9

    Article  MathSciNet  Google Scholar 

  24. Saadeh R, Ghazal B, Burqan A (2023) A study of double general transform for solving fractional partial differential equations. Math Methods Appl Sci 46(16):17158–17176

    Article  MathSciNet  Google Scholar 

  25. Matouk AE (2023) Chaotic attractors that exist only in fractional-order case. J Adv Res 45:183–192

    Article  CAS  PubMed  Google Scholar 

  26. Hasan FL, Abdoon MA (2021) The generalized (2 + 1) and (3+1)-dimensional with advanced analytical wave solutions via computational applications. Int J Nonlinear Anal Appl 2021(2):1213–1241

    Google Scholar 

  27. Hosseini K, Mayeli P, Ansari R (2018) Bright and singular soliton solutions of the conformable time-fractional Klein–Gordon equations with different nonlinearities. Waves Random Complex Media 28(3):426–434

    Article  ADS  MathSciNet  Google Scholar 

  28. Abdoon MA, Hasan FL (2022) Advantages of the differential equations for solving problems in mathematical physics with symbolic computation. Math Model Eng Probl 09(01):268–276

    Article  Google Scholar 

  29. Qazza A, Saadeh R (2023) On the analytical solution of fractional SIR epidemic model. Appl Comput Intell Soft Comput 2023:1–16

    Google Scholar 

  30. Elbadri M, Abdoon MA, Berir M, Almutairi DK (2023) A numerical solution and comparative study of the symmetric rossler attractor with the generalized Caputo fractional derivative via two different methods. Mathematics 11(13):2997

    Article  Google Scholar 

  31. Kumar S, Ahmadian A, Kumar R, Kumar D, Singh J, Baleanu D, Salimi M (2020) An efficient numerical method for fractional SIR epidemic model of infectious disease by using Bernstein wavelets. Mathematics 8(4):558

    Article  Google Scholar 

  32. Taghvaei A, Georgiou TT, Norton L, Tannenbaum A (2020) Fractional SIR epidemiological models. Sci Rep 10(1):20882

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  33. Losada J, Nieto JJ (2015) Properties of a new fractional derivative without singular kernel. Progr Fract Differ Appl 1(2):558

    Google Scholar 

  34. Chua LO, Kang SM (1976) Memristive devices and systems. Proc IEEE Inst Electr Electron Eng 64(2):209–23

    Article  MathSciNet  Google Scholar 

  35. Muthuswamy B, Chua LO (2010) Simplest chaotic circuit. Int J Bifurcat Chaos Appl Sci Eng 20(05):1567–1580

    Article  Google Scholar 

  36. Sah MP, Yang C, Kim H, Muthuswamy B, Jevtic J, Chua L (2015) A generic model of memristors with parasitic components. IEEE Trans Circuits Syst I 62(3):891–898

    Article  MathSciNet  Google Scholar 

  37. Hu H, Cao Y, Xu J, Ma C, Yan H (2021) An image compression and encryption algorithm based on the fractional-order simplest chaotic circuit. IEEE Access 9:22141–22155

    Article  Google Scholar 

  38. Ruan J, Sun K, Mou J, He S, Zhang L (2018) Fractional-order simplest memristor-based chaotic circuit with new derivative. Eur Phys J Plus 133(1):3

    Article  Google Scholar 

  39. Teng L, Iu HHC, Wang X, Wang X (2014) Chaotic behavior in fractional-order memristor-based simplest chaotic circuit using fourth degree polynomial. Nonlinear Dyn 77(1–2):231–241

    Article  Google Scholar 

  40. Jafari H, Khalique C, Nazari M (2011) Application of the Laplace decomposition method for solving linear and nonlinear fractional diffusion-wave equations. Appl Math Lett 24:1799–1805

    Article  MathSciNet  Google Scholar 

  41. Wazwaz AM (2010) Partial differential equations and solitary waves theory. Springer, Berlin

    Google Scholar 

  42. Toufik M, Atangana A (2017) New numerical approximation of fractional derivative with non-local and non-singular kernel: application to chaotic models. Eur Phys J Plus 132:444

    Article  Google Scholar 

  43. Bhalekar S, Daftardar-Geji V (2012) Solving a system of nonlinear functional equations using revised new iterative method. Int J Math Comput Phys Electron Comput Eng 6:8–21

    Google Scholar 

  44. Qazza A, Abdoon M, Saadeh R, Berir M (2023) A new scheme for solving a fractional differential equation and a chaotic system. Eur J Pure Appl Math 2023(2):1128–1139

    Article  Google Scholar 

  45. Saadeh R, Abdoon M, Qazza A, Berir M (2023) A numerical solution of generalized Caputo fractional initial value problems. Fractal Fract 7(4):332

    Article  Google Scholar 

  46. Elbadri M, Abdoon MA, Berir M, Almutairi DK (2023) A symmetry chaotic model with fractional derivative order via two different methods. Symmetry 15(6):1151

    Article  ADS  Google Scholar 

  47. Qazza A, Saadeh R, Alayed O, El-Ajou A (2023) Effective transform-expansions algorithm for solving non-linear fractional multi-pantograph system. AIMS Math 8(9):19950–19970

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors would like to extend their sincere appreciation to the Researchers Supporting Project number (RSPD2023R920), King Saud University, Saudi Arabia.

Funding

The Researchers Supporting Project Number (RSPD2023R920), King Saud University, Saudi Arabia.

Author information

Authors and Affiliations

  1. Department of Mathematics, College of Science, King Saud University, P.O. Box 1142, 11989, Riyadh, Saudi Arabia

    Abdulrahman B. M. Alzahrani

  2. Department of Mathematics, Zarqa University, Zarqa, 13110, Jordan

    Rania Saadeh & Ahmad Qazza

  3. Department of Basic Sciences, Common First Year Deanship, King Saud University, P.O. Box 1142, 12373, Riyadh, Saudi Arabia

    Mohamed A. Abdoon

  4. Department of Mathematics, Faculty of Sciences and Arts, Jouf University, Tubarjal, Saudi Arabia

    Mohamed Elbadri

  5. Department of Mathematic, University of Gezira, Wad Madani, Sudan

    Mohamed Elbadri

  6. Department of Mathematics, Faculty of Science and Arts, Al-Baha University, 1988, Baljurashi, Saudi Arabia

    Mohammed Berir

Authors
  1. Abdulrahman B. M. Alzahrani
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Rania Saadeh
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Mohamed A. Abdoon
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Mohamed Elbadri
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Mohammed Berir
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. Ahmad Qazza
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

Author Contributions: Conceptualization: RS, MA, ME and AQ, methodology: AA, RS, MB and AQ, software: AA, RS, MB and AQ, formal analysis: RS, MA, MB and AQ, investigation: MA, ME, MB & AQ, resources: AA, RS, ME, MB, data curation AA, ME, MB and AQ, writing—original draft preparation: AA, RS and AQ, writing—review and editing: AA, RS, MA, ME, MB, and AQ. All authors reviewed the manuscript.

Corresponding author

Correspondence to Ahmad Qazza.

Ethics declarations

Conflict of interest

The authors declare no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Alzahrani, A.B.M., Saadeh, R., Abdoon, M.A. et al. Effective methods for numerical analysis of the simplest chaotic circuit model with Atangana–Baleanu Caputo fractional derivative. J Eng Math 144, 9 (2024). https://doi.org/10.1007/s10665-023-10319-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10665-023-10319-x

Keywords

Search

Navigation