, 92:91 | Cite as

Dynamical analysis of a new three-dimensional fractional chaotic system

  • P Gholamin
  • A H Refahi SheikhaniEmail author


In the present paper, a new fractional chaotic system proposed by the authors is discussed. Moreover, based on the stability theory of fractional-order systems, conditions for the stability of nonlinear fractional-order systems are presented, and the existence and uniqueness of the solutions of the resulting new fractional chaotic attractor are also studied. Next, the necessary conditions for the existence of chaotic attractors in new fractional chaotic system are reported, and at the end the stability analysis of the corresponding equilibria is given. Last but not the least, the presented numerical simulations confirm the validity of our analysis.


New fractional chaotic system Grünwald–Letnikov derivative stability conditions fractional Hopf bifurcation numerical simulations 


02.30.Oz 05.45.Pq 95.10.Fh 


  1. 1.
    K B Oldham and J Spanier, The fractional calculus (Academic Press, New York, 1974)zbMATHGoogle Scholar
  2. 2.
    P J Torvik and R L Bagley, Trans. ASME 51, 294 (1984)CrossRefGoogle Scholar
  3. 3.
    R Hilfer, Applications of fractional calculus in physics (World Scientific, New Jersey, 2000)CrossRefGoogle Scholar
  4. 4.
    A E M El-Misiery and E Ahmed, Appl. Math. Comput. 178, 207 (2006)MathSciNetGoogle Scholar
  5. 5.
    F J V Parada, J A O Tapia and J A Ramirez, Physica A 373, 339 (2007)ADSCrossRefGoogle Scholar
  6. 6.
    Z D Mei, J G Peng and J H Gao, Indagationes Mathematicae 26, 669 (2015)Google Scholar
  7. 7.
    B Xin, T Chen and Y Liu, Commun. Nonlinear Sci. Numer. Simulat. 16, 4479 (2011)ADSMathSciNetCrossRefGoogle Scholar
  8. 8.
    W M Ahmad and R El-Khazali, Chaos Solitons Fractals 33, 1367 (2007)ADSMathSciNetCrossRefGoogle Scholar
  9. 9.
    H Aminikhah, A H Refahi Sheikhani and H Rezazadeh, Ain Shams Eng. J. (2016), CrossRefGoogle Scholar
  10. 10.
    H Rezazadeh, H Aminikhah and A H Refahi Sheikhani, Math. Commun. 21, 45 (2016)MathSciNetGoogle Scholar
  11. 11.
    F Mehrdoust, A H Refahi Sheikhani, M Mashoof and S Hasanzadeh, J. Econom. Studies 44(3), 489 (2017)CrossRefGoogle Scholar
  12. 12.
    M Mashoof and A H Refahi Sheikhani, U.P.B. Sci. Bull. Ser. A 79, 193 (2017)Google Scholar
  13. 13.
    M Axtell and E M Bise, Proc. of the IEEE Nat. Aerospace and Electronics Conference (New York, USA, 1990)Google Scholar
  14. 14.
    I Podlubny, IEEE Trans. Autom. Control 44, 208 (1999)CrossRefGoogle Scholar
  15. 15.
    P Arena, R Caponetto, L Fortuna and D Porto, Nonlinear noninteger order circuits and systems – An introduction (World Scientific, Singapore, 2000)CrossRefGoogle Scholar
  16. 16.
    S Westerlund, Dead matter has memory!, Causal consulting (Kalmar, Sweden, 2002)Google Scholar
  17. 17.
    B M Vinagre, Y Q Chen and I Petráš, J. Frankl. Inst. 340, 349 (2003)CrossRefGoogle Scholar
  18. 18.
    R L Magin, Fractional calculus in bioengineering (Begell House Publishers, Redding, 2006)Google Scholar
  19. 19.
    L M Wang, Pramana – J. Phys. 89: 38 (2017)ADSCrossRefGoogle Scholar
  20. 20.
    K Rabah, S Ladaci and M Lashab, Pramana – J. Phys. 89: 46 (2017)ADSCrossRefGoogle Scholar
  21. 21.
    R L Bagley and R A Calico, J. Guid. Control Dyn. 14, 304 (1991)ADSCrossRefGoogle Scholar
  22. 22.
    M Ichise, Y Nagayanagi and T Kojima, J. Electroanal. Chem. 33, 253 (1971)CrossRefGoogle Scholar
  23. 23.
    H H Sun, A A Abdelwahad and B Onaral, IEEE Trans. Automat. Control 29, 441 (1984)CrossRefGoogle Scholar
  24. 24.
    O Heaviside, Electromagnetic theory (Chelsea, New York, 1971)zbMATHGoogle Scholar
  25. 25.
    N Laskin, Physica A 287, 482 (2000)ADSMathSciNetCrossRefGoogle Scholar
  26. 26.
    D Kusnezov, A Bulgac and G D Dang, Phys. Rev. Lett. 82, 1136 (1999)ADSCrossRefGoogle Scholar
  27. 27.
    I Grigorenko and E Grigorenko, Phys. Rev. Lett. 91, 034101 (2003)ADSCrossRefGoogle Scholar
  28. 28.
    C G Li and G R Chen, Physica A 341, 55 (2004)ADSMathSciNetCrossRefGoogle Scholar
  29. 29.
    D Cafagna and G Grassi, Int. J. Bifurc. Chaos 18, 1845 (2008)CrossRefGoogle Scholar
  30. 30.
    H Zhu, S B Zhou and J Zhang, Chaos Solitons Fractals 39, 1595 (2009)ADSCrossRefGoogle Scholar
  31. 31.
    I Petráš, IEEE Trans. Circuits Syst. II, Express Briefs 57, 975 (2010)CrossRefGoogle Scholar
  32. 32.
    J G Lu, Phys. Lett. A 354, 305 (2006)ADSCrossRefGoogle Scholar
  33. 33.
    X Y Wang and M J Wang, Chaos 17, 033106 (2007)ADSCrossRefGoogle Scholar
  34. 34.
    H Aminikhah, A H Refahi Sheikhani and H Rezazadeh, Sci. World J. 2013, 1 (2013)Google Scholar
  35. 35.
    P Gholamin and A H Refahi Sheikhani, Chin. J. Phys. 55, 1300 (2017)CrossRefGoogle Scholar
  36. 36.
    I Podlubny, Fractional differential equations (Academic Press, San Diego, 1999)zbMATHGoogle Scholar
  37. 37.
    A A Kilbas, H M Srivastava and J J Trujillo, Theory and applications of fractional differential equations (Elsevier, Amsterdam, The Netherlands, 2006)zbMATHGoogle Scholar
  38. 38.
    K Diethelm and N J Ford, J. Math. Anal. Appl. 265, 229 (2002)MathSciNetCrossRefGoogle Scholar
  39. 39.
    M S Tavazoei and M Haeri, Physica D 237, 2628 (2008)ADSMathSciNetCrossRefGoogle Scholar
  40. 40.
    M S Tavazoei and M Haeri, IET Signal Proc. 1, 171 (2007)CrossRefGoogle Scholar
  41. 41.
    M S Tavazoei and M Haeri, Phys. Lett. A 367, 102 (2007)ADSCrossRefGoogle Scholar
  42. 42.
    I Petráš, Nonlinear Dyn. 57, 157 (2009)ADSCrossRefGoogle Scholar
  43. 43.
    K Diethelm, N J Ford, A D Freed and Y Luchko, Comput. Methods Appl. Mech. Eng. 194, 743 (2005)ADSCrossRefGoogle Scholar
  44. 44.
    E Ahmed, A M A El-Sayed and H A A El-Saka, Phys. Lett. A 358, 1 (2006)ADSMathSciNetCrossRefGoogle Scholar
  45. 45.
    J Čermák and L Nechvátal, Nonlinear Dyn. 87, 939 (2017)CrossRefGoogle Scholar
  46. 46.
    H K Khalil, Nonlinear systems, 3rd edn (Prentice-Hall, 1992)Google Scholar
  47. 47.
    H Saberi Najafi, S A Edalatpanah and A H Refahi Sheikhani, Mediterr. J. Math. 11, 1019 (2014)MathSciNetCrossRefGoogle Scholar
  48. 48.
    H Saberi Najafi and A H Refahi Sheikhani, Appl. Math. Comput. 184, 421 (2007)MathSciNetGoogle Scholar
  49. 49.
    L O Chua, M Komuro and T Matsumoto, IEEE Trans. Circ. Syst. 33, 1072 (1986)Google Scholar
  50. 50.
    A S Deshpande, V Daftardar-Gejji and Y V Sukale, Chaos Solitons Fractals 98, 189 (2017)ADSMathSciNetCrossRefGoogle Scholar

Copyright information

© Indian Academy of Sciences 2019

Authors and Affiliations

  1. 1.Department of Applied Mathematics, Faculty of Mathematical Sciences, Lahijan BranchIslamic Azad UniversityLahijanIran

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