Time evolution of nonadditive entropies: The logistic map

Due to the second principle of thermodynamics, time dependence entropy for all kinds systems under 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 freedom, nor closed, dissipative. It exhibits, nevertheless, a peculiar evolution its natural entropy, which additive Boltzmann-Gibbs-Shannon one, $S_{BG}=-\sum_{i=1}^W p_i \ln p_i$, values $a$ Lyapunov exponent positive, and nonadditive $S_q= \frac{1-\sum_{i=1}^W p_i^q}{q-1}$ with $q=0.2445\dots$ at edge chaos, where vanishes, $W$ being number windows phase space partition. We numerically show that, increasing time, phase-space-averaged overshoots above stationary-state value in cases. However, when $W\to\infty$, overshooting gradually disappears most chaotic case ($a=2$), whereas, remarkable contrast, appears monotonically diverge Feigenbaum point ($a=1.4011\dots$). Consequently, achieved from {\it above}, instead below}, as could have been priori expected. These results raise question whether usual requirements -- generic initial conditions validity might be necessary but not sufficient.