$\eta=1/2$ to be a good threshold to separate order from chaos. For the stochastic dynamics on small-world networks we choose as the threshold the approximate value of the entropy at the first bifurcation as shown in the figures. For values smaller than this threshold there are noisy periodic orbits. The bifurcation diagram of the figure is reminiscent of the mean-field one, Fig. \[fig:mf-J\].

Notice that pitchfork bifurcations (dependence on the initial magnetization) are present also in the microscopic simulations, as shown in Fig. \[bifp\]

In Fig. \[fig:sw-J\] we show the bifurcation diagrams and entropy of the magnetization $m$ for the small-world networks obtained for different values of $p$. As before, the entropy is a good indicator of disorder.

![\[bifp\] (Color online) Small-world stochastic bifurcation diagram of the magnetization $m$, dots (in magenta), and the entropy $\eta$, continuous curve (in green), as functions of the long-range connection probability $p$ with $J=-10$. The “jump” of $m$ for $p\simeq 0.45$ corresponds to a pitchfork bifurcation (dependence on the initial magnetization).](sw-bif-eta-PI-1-o2-N10000-k20-J-10p00-W15p00-c0p50){width="\columnwidth"}

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                                                                                                                                                                                        $p=0.3$                                                                                                                                                                                                                                                                                                                                                                              $p=0.5$
   ![\[fig:sw-J\] (Color online) Small-world bifurcation diagram of the magnetization $m$, dots (in magenta) and the corresponding entropy $\eta$, continuous line (in green), as functions of of the linear coupling 