On periodic solutions of fractional-order differential systems with a fixed length of sliding memory

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Safa Bourafa
Mohammed Salah Abdelouahab
René Lozi


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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Bourafa, S., Abdelouahab, M.S. and Lozi, R. 2021. On periodic solutions of fractional-order differential systems with a fixed length of sliding memory. Journal of Innovative Applied Mathematics and Computational Sciences. 1, 1 (Dec. 2021), 64–78.
Research Articles


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