  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.
