  We present some distinct asymptotic properties of solutions to Caputo
fractional differential equations (FDEs). First, we show that the non-trivial
solutions to a FDE can not converge to the fixed points faster than
$t^{-\alpha}$, where $\alpha$ is the order of the FDE. Then, we introduce the
notion of Mittag-Leffler stability which is suitable for systems of
fractional-order. Next, we use this notion to describe the asymptotical
behavior of solutions to FDEs by two approaches: Lyapunov's first method and
Lyapunov's second method. Finally, we give a discussion on the relation between
Lipschitz condition, stability and speed of decay, separation of trajectories
to scalar FDEs.
