  In this paper, we study 1D autonomous fractional ODEs $D_c^{\gamma}u=f(u), 0<
\gamma <1$, where $u: [0,\infty)\mapsto\mathbb{R}$ is the unknown function and
$D_c^{\gamma}$ is the generalized Caputo derivative introduced by Li and Liu (
arXiv:1612.05103). Based on the existence and uniqueness theorem and regularity
results in previous work, we show the monotonicity of solutions to the
autonomous fractional ODEs and several versions of comparison principles. We
also perform a detailed discussion of the asymptotic behavior for $f(u)=Au^p$.
In particular, based on an Osgood type blow-up criteria, we find relatively
sharp bounds of the blow-up time in the case $A>0, p>1$. These bounds indicate
that as the memory effect becomes stronger ($\gamma\to 0$), if the initial
value is big, the blow-up time tends to zero while if the initial value is
small, the blow-up time tends to infinity. In the case $A<0, p>1$, we show that
the solution decays to zero more slowly compared with the usual derivative.
Lastly, we show several comparison principles and Gr\"onwall inequalities for
discretized equations, and perform some numerical simulations to confirm our
analysis.
