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

Safa Bourafa , René Lozi (1) , Mohammed-Salah Abdelouahab (2)

1 LJAD - Laboratoire Jean Alexandre Dieudonné
2 Laboratoire de Mathématiques et leurs interactions, centre universitaire de Mila

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.

Mots clés: Dynamical system; Fractional-order derivative; sliding fixed memory length; periodic solution

Domaines: Systèmes dynamiques [math.DS]

Safa Bourafa, René Lozi, Mohammed-Salah Abdelouahab. On periodic solutions of fractional-order differential systems with a fixed length of sliding memory. Journal of Innovative Applied Mathematics and Computational Sciences, 2021, 1 (1), pp.64-78. ⟨hal-03513190⟩