there is always a stochastic noise that increases the value of the entropy in the “fixed-point” part of the parameter space. This base-level value is related to the size of the sample, and slowly vanishes for large samples.

In order to use finite-size samples, we set the onset of the phase in the stochastic systems corresponding to the chaotic phase in the deterministic ones to the mid-value of the range of $\eta$. Taking the limits $T\to\infty$, $L\to\infty$ leads to the Kolmogorov-Sinai entropy [@kolmogorov58; @kolmogorov59; @sinai59; @ott02].

Before presenting the different scenarios, let us illustrate the type of bifurcations that are present. In Figs. \[fig:mf-bif-c0p60\] (a) and (b) we show parts of the bifurcation diagram of the map of Eq.  as a function of $J$ staring with different values of the initial magnetization $m_0$.

Referring to the values in the Figure, at $J=J_0$ there is a pitchfork bifurcation, i.e., a separation of basins, that reunite at $J=J_1$, which is another pitchfork bifurcation, in the reverse direction. Intermixed, there are period-doubling bifurcations. There are other pitchfork bifurcations for different intervals of $J$.

In Fig. \[fig:mf-bif-c0p60\] (c) we show the return map of the mean-field map for $J=-5.2$. We can see that there 