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The fractional-order derivative of a non-constant periodic function is not periodic with the same period. Consequently, any time-invariant fractional-order systems do not have a non-constant periodic solution. This property limits the applicability of fractional derivatives and makes it unfavorable to model periodic real phenomena.
This article introduces a modification to the Caputo and Rieman-Liouville fractional-order operators by fixing their memory length and varying the lower terminal. It is shown that this modified definition of fractional derivative preserves the periodicity. Therefore, periodic solutions can be expected in fractional-order systems in terms of the new fractional derivative operator. To confirm this assertion, one investigates two examples, one linear system for which one gives an exact periodic solution by its analytical expression and another nonlinear system for which one provides exact periodic solutions using qualitative and numerical methods.
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M-S. Abdelouahab and N. Hamri, The Grünwald-Letnikov fractional-order derivative with fixed memory length, Mediterr. J. Math. 13 (2016), 557–572.
M-S. Abdelouahab, N. Hamri and J. W. Wang, Hopf bifurcation and chaos in fractionalorder modified hybrid optical system, Nonlinear Dyn. 69 (2012), 275–284.
M-S. Abdelouahab and R. Lozi, Hopf Bifurcation and Chaos in Simplest Fractional-Order Memristor-based Electrical Circuit, Indian. J. Ind. Appl. Math. 6(2) (2015), 105–119.
M-S. Abdelouahab, R. Lozi and G. Chen, Complex Canard Explosion in a Fractional-Order FitzHugh–Nagumo Model, Int. J. Bifurcation Chaos. 29(8) (2019), 1950111–1950133.
I. Area, J. Losada and J. J. Nieto, On quasi-periodicity properties of fractional integrals and fractional derivatives of periodic functions, Integral Transforms Spec. Funct. 27 (2016), 1–16.
R. L. Bagley and R. A. Calico, Fractional order state equations for the control of viscoelastically damped structures, J. Guid. Control Dyn. 14 (1991), 304–311.
S. Bourafa, M-S. Abdelouahab and A. Moussaoui, On some extended Routh-Hurwitz conditions for fractional-order autonomous systems of order a 2 [0, 2) and their applications to
some population dynamic models, Chaos Solitons Fractals. 133 (2020), 109623.  P. L. Butzer and U. Westphal, An introduction to fractional calculus, In: Hilfer, R. Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000.
M. Caputo, Linear models of dissipation whose Q is almost frequency independent-II, Geophys. J. R. Astron. Soc. 13 (1967), 529–539.
O. Heaviside, H. J. Josephs and B. A. Behrend, Electromagnetic Theory, Chelsea Publishing Company, New York, 1971.
M. Ichise, Y. Nagayanagi and T. Kojima, An analog simulation of noninteger order transfer functions for analysis of electrode process, J. Electroanal. Chem. 33 (1971), 253–265.
Y-M. Kang, Y. Xie, J-C. Lu and J. Jiang, On the nonexistence of non-constant exact periodic solutions in a class of the Caputo fractional-order dynamical systems, Nonlinear Dyn. 82 (2015), 1259–1267.
E. Kaslik and S. Sivasundaram, Non-existence of periodic solutions in fractional-order dynamical systems and a remarkable difference between integer and fractional-order derivatives of periodic functions, Nonlinear Analysis: Real World Applications. 13 (2012), 1489–1497.
D. Kusnezov, A. Bulgac and G. D. Dang, Quantum levy processes and fractional kinetics, Phys. Rev. Lett. 82 (1999), 1136–1139.
C. A. Monje, Y. Q. Chen, B. M. Vinagre, D. Xue, and V. Feliu, Fractional-Order Systems and Controls Fundamentals and Applications, Springer-Verlag London Limited, London, 2010.
K. B. Oldham and J. Spanier, The Fractional Calculus Theory and Applications of Differentiation and Integration to Arbitrary Order, Academic press, inc, USA, 1974.
I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.
B. Ross, The development of fractional calculus 1695-1900, Hist. Math. 4 (1977), 75–89.
S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, Amsterdam, 1993.
H. H. Sun, A. A. Abdelwahab and B. Onaral, Linear approximation of transfer function with a pole of fractional order, IEEE Trans. Autom. Control. 29 (1984), 441–444.
J. T. Tanner, The stability and the intrinsic growth rates of prey and predator populations, Ecology. 56 (1975), 855–867.
M. S. Tavazoei, A note on fractional-order derivatives of periodic functions. Automatica J. IFAC. 46 (2010), 945–948.
M. S. Tavazoei and M. Haeri, A proof for non existence of periodic solutions in time invariant fractional order systems, Automatica J. IFAC. 45 (2009), 1886–1890.