case with fixed connectivity $K$, follows from the Markov equation assuming no spatial correlations. Then $$\begin{aligned}
\label{eq:mf}
 m'=&f(m) = \frac{1}{2^{K}}\sum_{k=0}^K\binom{K}{k}(1+m)^k(1-m)^{K-k}\nonumber\\
 &\times\tanh\left[J\left(\frac{2k}{K}-1\right)+W\left(\frac{2k}{K}-1\right)^3\right],\end{aligned}$$ with $m=m(t)$ and $m'=m(t+1)$. We show in Fig. \[fig:tau\] (b) the graph of $m'$ together with some iterates of the map.

The mean-field magnetization exhibits chaos that can be characterized by the Lyapunov exponent $\lambda$ [@ott02]. However, on spatially extended networks $m$ changes stochastically and cannot be characterized in the same way.

In order to compare microscopic and mean-field models within the same framework we use the Boltzmann’s entropy $\eta$ [@Boltzmann; @bagnoli2013] of the magnetization $m$. The interval $[-1,1]$ is partitioned in $L$ disjoint intervals $I_i$ of equal size and the probability $q_i$ of $I_i$ is the fraction of visits to $I_i$ after $T$ time steps with $T\gg 1$.

Once these probabilities are known, $\eta$ is defined by $$\label{eq:s}
 {\eta}=- \dfrac{1}{\log L}\sum_{i=1}^Lq_i\log q_i,$$ so that $0\leq \eta\leq 1$, the lower bound corresponding to a fixed point, the upper one to the uniform distribution $q_i=1/L$.

For a periodic orbit of period $2^p$ and $L=2^b$, $\eta=p/b$. For low-dimensional dynamical systems, like the mean-field equation, the mid-value threshold $\eta=0.5$ effectively separates the contracting dynamics (cycles) from the chaotic ones. For spatially-extended systems, 