Non-existence of periodic solutions in fractional-order dynamical systems and a remarkable difference between integer and fractional-order derivatives of periodic functions

TL;DR

This paper demonstrates that fractional-order derivatives of periodic functions cannot be periodic, leading to the non-existence of exact periodic solutions in many fractional-order dynamical systems, contrasting with integer-order cases.

Contribution

It proves the non-existence of exact periodic solutions in fractional-order systems using Mellin transform and analyzes implications for neural network models.

Findings

Fractional derivatives of periodic functions are not periodic.

Exact periodic solutions do not exist in a broad class of fractional systems.

Numerical limit cycles in fractional neural networks are not truly periodic.

Abstract

Using the Mellin transform approach, it is shown that, in contrast with integer-order derivatives, the fractional-order derivative of a periodic function cannot be a function with the same period. The three most widely used definitions of fractional-order derivatives are taken into account, namely, the Caputo, Riemann-Liouville and Grunwald-Letnikov definitions. As a consequence, the non-existence of exact periodic solutions in a wide class of fractional-order dynamical systems is obtained. As an application, it is emphasized that the limit cycle observed in numerical simulations of a simple fractional-order neural network cannot be an exact periodic solution of the system.