  Due to the second principle of thermodynamics, the time dependence of entropy
for all kinds of systems under all kinds of physical circumstances always
thrives interest. The logistic map $x_{t+1}=1-a x_t^2 \in [-1,1]\;(a\in [0,2])$
is neither large, since it has only one degree of freedom, nor closed, since it
is dissipative. It exhibits, nevertheless, a peculiar time evolution of its
natural entropy, which is the additive Boltzmann-Gibbs-Shannon one,
$S_{BG}=-\sum_{i=1}^W p_i \ln p_i$, for all values of $a$ for which the
Lyapunov exponent is positive, and the nonadditive one $S_q=
\frac{1-\sum_{i=1}^W p_i^q}{q-1}$ with $q=0.2445\dots$ at the edge of chaos,
where the Lyapunov exponent vanishes, $W$ being the number of windows of the
phase space partition. We numerically show that, for increasing time, the
phase-space-averaged entropy overshoots above its stationary-state value in all
cases. However, when $W\to\infty$, the overshooting gradually disappears for
the most chaotic case ($a=2$), whereas, in remarkable contrast, it appears to
monotonically diverge at the Feigenbaum point ($a=1.4011\dots$). Consequently,
the stationary-state entropy value is achieved from {\it above}, instead of
from {\it below}, as it could have been a priori expected. These results raise
the question whether the usual requirements -- large, closed, and for generic
initial conditions -- for the second principle validity might be necessary but
not sufficient.
