Skip to main content
SpringerLink
Log in

A highly chaotic fractional-order system with a four-wing attractor and its synchronization

  • Published:
International Journal of Dynamics and Control Aims and scope Submit manuscript

Abstract

In this paper, a new four-wing attractor is reported in a fractional-order chaotic system. The chaoticness of the proposed system is investigated by obtaining Lyapunov exponents and compared with that of well-known chaotic systems in the literature. The findings reveal extremely high chaoticness of the introduced system which makes it a proper choice for encryption systems and secure communication. Furthermore, synchronization of the proposed system is studied in this paper. Sliding mode control has been used for this purpose and it is proven and illustrated that the synchronization error is asymptotically stable by employing the Lyapunov stability theorem.

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
Fig. 4
Fig. 5
Fig. 6
Fig. 7

Similar content being viewed by others

Availability of data and material

Not applicable to this article as no datasets were generated or analysed during the current study.

Code availability

Codes have been written in MATLAB and are available on reasonable request.

References

  1. Cao J, Chen Y, Wang Y, Cheng G, Barrière T (2020) Shifted Legendre polynomials algorithm used for the dynamic analysis of PMMA viscoelastic beam with an improved fractional model. Chaos Solitons Fract 141:110342

    Article  MathSciNet  Google Scholar 

  2. Long Y, Xu B, Chen D, Ye W (2018) Dynamic characteristics for a hydro-turbine governing system with viscoelastic materials described by fractional calculus. Appl Math Model 58:128–139

    Article  MathSciNet  Google Scholar 

  3. Yang XJ, Abdel-Aty M, Cattani C (2019) A new general fractional-order derivataive with Rabotnov fractional-exponential kernel applied to model the anomalous heat transfer. Therm Sci 23(3 Part A):1677–1681

    Article  Google Scholar 

  4. Wei T, Li YS (2018) Identifying a diffusion coefficient in a time-fractional diffusion equation. Math Comput Simul 151:77–95

    Article  MathSciNet  Google Scholar 

  5. Wu GZ, Yu LJ, Wang YY (2020) Fractional optical solitons of the space-time fractional nonlinear Schrödinger equation. Optik 207:164405

    Article  Google Scholar 

  6. Sene N (2020) SIR epidemic model with Mittag-Leffler fractional derivative. Chaos, Solitons Fract 137:109833

    Article  MathSciNet  Google Scholar 

  7. Dokuyucu MA, Dutta H (2020) A fractional order model for Ebola Virus with the new Caputo fractional derivative without singular kernel. Chaos Solitons Fract 134:109717

    Article  MathSciNet  Google Scholar 

  8. Rajagopal K, Hasanzadeh N, Parastesh F, Hamarash II, Jafari S, Hussain I (2020) A fractional-order model for the novel coronavirus (COVID-19) outbreak. Nonlinear Dyn 101(1):711–718

    Article  Google Scholar 

  9. Hafezi A, Khandani K, Majd VJ (2020) Non-fragile exponential polynomial observer design for a class of nonlinear fractional-order systems with application in chaotic communication and synchronisation. Int J Syst Sci 51(8):1353–1372

    Article  MathSciNet  Google Scholar 

  10. Khandani K, Majd VJ, Tahmasebi M (2016) Robust stabilization of uncertain time-delay systems with fractional stochastic noise using the novel fractional stochastic sliding approach and its application to stream water quality regulation. IEEE Trans Autom Control 62(4):1742–1751

    Article  MathSciNet  Google Scholar 

  11. Tarasov VE, Tarasova VV (2018) Macroeconomic models with long dynamic memory: fractional calculus approach. Appl Math Comput 338:466–486

    MathSciNet  MATH  Google Scholar 

  12. Zouad F, Kemih K, Hamiche H (2019) A new secure communication scheme using fractional order delayed chaotic system: design and electronics circuit simulation. Analog Integr Circ Sig Process 99(3):619–632

    Article  Google Scholar 

  13. Bettayeb M, Al-Saggaf UM, Djennoune S (2018) Single channel secure communication scheme based on synchronization of fractional-order chaotic Chua’s systems. Trans Inst Meas Control 40(13):3651–3664

    Article  Google Scholar 

  14. Danca MF, Kuznetsov N (2018) Matlab code for Lyapunov exponents of fractional-order systems. Int J Bifurc Chaos 28(05):1850067

    Article  MathSciNet  Google Scholar 

  15. Podlubny I (1998) Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications. Elsevier

    MATH  Google Scholar 

  16. ] Li, Y., Chen, Y., & Podlubny, I. (2009) Mittag-Leffler stability of fractional order nonlinear dynamic systems. Automatica 45(8):1965–1969

    Article  MathSciNet  Google Scholar 

  17. Li C, Deng W (2007) Remarks on fractional derivatives. Appl Math Comput 187(2):777–784

    MathSciNet  MATH  Google Scholar 

  18. Signing VF, Kengne J, Kana LK (2018) Dynamic analysis and multistability of a novel four-wing chaotic system with smooth piecewise quadratic nonlinearity. Chaos Solitons Fract 113:263–274

    Article  MathSciNet  Google Scholar 

  19. Odibat Z, Momani S, Erturk VS (2008) Generalized differential transform method: Application to differential equations of fractional order. Appl Math Comput 197(2):467–477

    MathSciNet  MATH  Google Scholar 

  20. Deng J, Ma L (2010) Existence and uniqueness of solutions of initial value problems for nonlinear fractional differential equations. Appl Math Lett 23(6):676–680

    Article  MathSciNet  Google Scholar 

  21. Lakshmikantham V, Vatsala AS (2008) General uniqueness and monotone iterative technique for fractional differential equations. Appl Math Lett 21(8):828–834

    Article  MathSciNet  Google Scholar 

  22. ur Rehman M, Khan RA (2010) Existence and uniqueness of solutions for multi-point boundary value problems for fractional differential equations. Appl Math Lett 23(9):1038–1044

    Article  MathSciNet  Google Scholar 

  23. Atta AG, Moatimid GM, Youssri YH (2019) Generalized Fibonacci operational collocation approach for fractional initial value problems. Int J Appl Comput Math 5(1):1–11

    Article  MathSciNet  Google Scholar 

  24. Lakshmikantham V, Vatsala AS (2008) Basic theory of fractional differential equations. Nonlinear Anal Theory Methods Appl 69(8):2677–2682

    Article  MathSciNet  Google Scholar 

  25. Hafez RM, Youssri YH (2020) Legendre-collocation spectral solver for variable-order fractional functional differential equations. Comput Methods Differ Equ 8(1):99–110

    MathSciNet  MATH  Google Scholar 

  26. Youssri YH, Abd-Elhameed WM, Mohamed AS, Sayed SM (2021) Generalized Lucas polynomial sequence treatment of fractional pantograph differential equation. Int J Appl Comput Math 7(2):1–16

    Article  MathSciNet  Google Scholar 

  27. Abd-Elhameed WM, Machado JAT, Youssri YH (2021) Hypergeometric fractional derivatives formula of shifted Chebyshev polynomials: Tau algorithm for a type of fractional delay differential equations. Int J Nonlinear Sci Numer Simul

  28. Hafez RM, Youssri YH (2020) Shifted Jacobi collocation scheme for multidimensional time-fractional order telegraph equation. Iran J Numer Anal Optim 10(1):195–223

    MATH  Google Scholar 

  29. Qi G, Chen G, Zhang Y (2008) On a new asymmetric chaotic system. Chaos Solitons Fract 37(2):409–423

    Article  Google Scholar 

  30. Singh JP, Roy BK (2018) A more chaotic and easily hardware implementable new 3-D chaotic system in comparison with 50 reported systems. Nonlinear Dyn 93(3):1121–1148

    Article  Google Scholar 

  31. Rössler OE (1976) An equation for continuous chaos. Phys Lett A 57(5):397–398

    Article  Google Scholar 

  32. Chen G, Ueta T (1999) Yet another chaotic attractor. Int J Bifurc Chaos 9(07):1465–1466

    Article  MathSciNet  Google Scholar 

  33. Bovy J (2004) Lyapunov exponents and strange attractors in discrete and continuous dynamical systems. Theor Phys Project Cathol Univ Leuven Flanders Belg Tech Rep 9:1–19

    Google Scholar 

  34. Li XF, Chlouverakis KE, Xu DL (2009) Nonlinear dynamics and circuit realization of a new chaotic flow: a variant of Lorenz, Chen and Lü. Nonlinear Anal Real World Appl 10(4):2357–2368

    Article  MathSciNet  Google Scholar 

  35. Lai Q, Huang J, Xu G (2016) Coexistence of multiple attractors in a new chaotic system. Acta Phys Pol B 47(10):2315–2323

    Article  MathSciNet  Google Scholar 

  36. Volos C, Akgul A, Pham VT, Stouboulos I, Kyprianidis I (2017) A simple chaotic circuit with a hyperbolic sine function and its use in a sound encryption scheme. Nonlinear Dyn 89(2):1047–1061

    Article  Google Scholar 

  37. Vaidyanathan S, Rajagopal K (2016) Analysis, control, synchronization and LabVIEW implementation of a seven-term novel chaotic system. Int J Control Theory Appl 9(1):151–174

    Google Scholar 

  38. Mohammadzadeh A, Kumbasar T (2020) A new fractional-order general type-2 fuzzy predictive control system and its application for glucose level regulation. Appl Soft Comput 91:106241

    Article  Google Scholar 

  39. Lin TC, Chen MC, Roopaei M (2011) Synchronization of uncertain chaotic systems based on adaptive type-2 fuzzy sliding mode control. Eng Appl Artif Intell 24(1):39–49

    Article  Google Scholar 

  40. Mohammadzadeh A, Hashemzadeh F (2015) A new robust observer-based adaptive type-2 fuzzy control for a class of nonlinear systems. Appl Soft Comput 37:204–216

    Article  Google Scholar 

Download references

Funding

No funding information.

Author information

Authors and Affiliations

  1. Department of Electrical Engineering, Faculty of Engineering, Arak University, Arak, Iran

    Mohammad Ebrahim Aghili & Khosro Khandani

  2. School of Electrical and Computer Engineering, Tarbiat Modares University, Tehran, Iran

    Majid Parvizian

Authors
  1. Mohammad Ebrahim Aghili
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Khosro Khandani
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Majid Parvizian
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

Conceptualization: Khosro Khandani; Methodology: Mohammad Ebrahim Aghili, Khosro Khandani, Majid Parvizian; Formal analysis and investigation: Mohammad Ebrahim Aghili, Khosro Khandani; Writing - original draft preparation: Mohammad Ebrahim Aghili, Khosro Khandani; Writing - review and editing: Mohammad Ebrahim Aghili, Khosro Khandani, Majid Parvizian; Software: Mohammad Ebrahim Aghili, Majid Parvizian; Supervision: Khosro Khandani.

Corresponding author

Correspondence to Khosro Khandani.

Ethics declarations

Conflict of interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Aghili, M.E., Khandani, K. & Parvizian, M. A highly chaotic fractional-order system with a four-wing attractor and its synchronization. Int. J. Dynam. Control 10, 1199–1207 (2022). https://doi.org/10.1007/s40435-021-00877-2

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40435-021-00877-2

Keywords

Search

Navigation