Mediterranean Journal of Mathematics

, Volume 13, Issue 2, pp 557–572 | Cite as

The Grünwald–Letnikov Fractional-Order Derivative with Fixed Memory Length

  • Mohammed-Salah Abdelouahab
  • Nasr-Eddine Hamri


Contrary to integer-order derivative, the fractional-order derivative of a non-constant periodic function is not a periodic function with the same period. As a consequence of this property, the time-invariant fractional-order systems do not have any non-constant periodic solution unless the lower terminal of the derivative is ±∞, which is not practical. This property limits the applicability of the fractional derivative and makes it unfavorable, for a wide range of periodic real phenomena. Therefore, enlarging the applicability of fractional systems to such periodic real phenomena is an important research topic. In this paper, we give a solution for the above problem by imposing a simple modification on the Grünwald–Letnikov definition of fractional derivative. This modification consists of fixing the memory length and varying the lower terminal of the derivative. It is shown that the new proposed definition of fractional derivative preserves the periodicity.

Mathematics Subject Classification

Primary 26A33 Secondary 33C20 


Fractional derivative memory length periodic function 


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  1. 1.
    Ross B.: The development of fractional calculus 1695–1900. Hist. Math. 4, 75–89 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Podlubny I.: Fractional Differential Equations. Academic Press, San Diego (1999)zbMATHGoogle Scholar
  3. 3.
    ldham K.B.O., Spanier J.: The Fractional Calculus Theory and Applications of Differentiation and Integration to Arbitrary Order. Academic press, inc, USA (1974)Google Scholar
  4. 4.
    Samko S.G., Kilbas A.A., Marichev O.I.: Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach, Amsterdam (1993)zbMATHGoogle Scholar
  5. 5.
    Butzer P.L., Westphal U.: An introduction to fractional calculus. In: Hilfer, R. (eds) Applications of Fractional Calculus in Physics, World Scientific, Singapore (2000)Google Scholar
  6. 6.
    Monje, C.A., Chen, Y.Q., Vinagre, B.M., Xue, D., Feliu, V.: Fractional-Order Systems and Controls Fundamentals and Applications. Springer-Verlag London Limited, London (2010)Google Scholar
  7. 7.
    Caputo M.: Linear models of dissipation whose Q is almost frequency independent-II. Geophys. J. R. Astron. Soc. 13, 529–539 (1967)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Bagley R.L., Calico R.A.: Fractional order state equations for the control of viscoelastically damped structures. J. Guid. Control Dyn. 14, 304–311 (1991)CrossRefGoogle Scholar
  9. 9.
    Sun H.H., Abdelwahab A.A., Onaral B.: Linear approximation of transfer function with a pole of fractional order. IEEE Trans. Autom. Control 29, 441–444 (1984)CrossRefzbMATHGoogle Scholar
  10. 10.
    Ichise M., Nagayanagi Y., Kojima T.: An analog simulation of noninteger order transfer functions for analysis of electrode process. J. Electroanal. Chem. 33, 253–265 (1971)CrossRefGoogle Scholar
  11. 11.
    Heaviside O.: Electromagnetic Theory. Chelsea, New York (1971)zbMATHGoogle Scholar
  12. 12.
    Kusnezov D., Bulgac A., Dang G.D.: Quantum levy processes and fractional kinetics. Phys. Rev. Lett. 82, 1136–1139 (1999)CrossRefGoogle Scholar
  13. 13.
    Abdelouahab, M.-S., Lozi, R., Chua, L.O.: Memfractance: a mathematical paradigm for circuit elements with memory. Int. J. Bifurc. Chaos 24(9), 28 p. (2014)Google Scholar
  14. 14.
    Abdelouahab, M.-S., Hamri, N., Wang, J.: Chaos Control of a Fractional-Order Financial System. Hindawi Pub Corp Math Prob in Engineering, pp. 1–18 (2010)Google Scholar
  15. 15.
    Abdelouahab M.-S., Hamri N.: Fractional-order hybrid optical system and its chaos control synchronization. EJTP 11(30), 49–62 (2014)Google Scholar
  16. 16.
    Miranda, J.G.: Synchronization and control of chaos: an introduction for scientists and engineers. Imperial College Press, London (2004)Google Scholar
  17. 17.
    Tavazoei M.S.: A note on fractional-order derivatives of periodic functions. Automatica 46, 945–948 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Tavazoei M.S., Haeri M.: A proof for non existence of periodic solutions in time invariant fractional order systems. Automatica 45, 1886–1890 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Tavazoei M.S., Haeri M., Attari M., Bolouki S., Siami M.: More details on analysis of fractional-order van der pol oscillator. J. Vib. Control 15(6), 803–819 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Yazdani M., Salarieh H.: On the existence of periodic solutions in time-invariant fractional order systems. Automatica 47, 1834–1837 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Kaslik E., Sivasundaram S.: Non-existence of periodic solutions in fractional-order dynamical systems and a remarkable difference between integer and fractional-order derivatives of periodic functions. Nonlinear Anal. Real World Appl. 13, 1489–1497 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Abdelouahab M.S., Hamri N., Wang J.W.: Hopf bifurcation and chaos in fractional-order modified hybrid optical system. Nonlinear Dyn. 69, 275–284 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Cafagna D., Grassi G.: On the simplest fractional-order memristor-based chaotic system. Nonlinear Dyn. 70, 1185–1197 (2012)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Diethelm K.: The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type. Springer, Berlin (2010)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer Basel 2015

Authors and Affiliations

  1. 1.Department of Mathematics and Computer SciencesMila University CentreMilaAlgeria

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