measure of chaos [@Boltzmann].

The scheme of the paper is the following. We discuss the “nonlinear” parallel Ising model in Section \[sec:parallelising\]. We can therefore introduce the mean-field approximation of the model in Section \[sec:meanfield\], showing the bifurcation phase diagrams as a function of the parameters. The definition of the entropy and the results of microscopic simulations $\eta$ are reported in Section \[sec:simulations\]. Finally, conclusions are drawn in the last Section. In this Section we discuss also the differences between the present and the original model of Refs. [@bagnoli2013; @bagnoli2015].

![\[plaquette\] (Color online.) A $K=3$ spin neighborhood, with the interaction terms corresponding to the external field $\tilde H$, the two-spin $\tilde J$, three-spin $\tilde Z$, and four-spin $\tilde W$ interaction constants.](plaquette){width="\columnwidth"}

Parallel nonlinear Ising model {#sec:parallelising}
==============================

We consider a system with $N$ sites, each one in a state $s_i\in
\{-1,1\}$, $i=1,\dots,N$. The state of the system is $@s=(s_1,\dots,s_{N})$. The topology of the system is defined by the adjacency matrix $a$ with $a_{ij}=1$ if site $j$ belongs to site $i$’s neighborhood and is zero otherwise. The connectivity $k_i$, the local field $\tilde{h}_i$, and the rescaled local field $h_i$ at site $i$ are $$k_i=\sum_j a_{ij},\qquad
  \tilde{h}_i=\sum_j a_{ij}s_j, \qquad
  h_i=\dfrac{\tilde{h}_i}{k_i},$$ with $h_i\in 