each one of its $K$ neighbors is replaced by a random one. Then the spin at each site is updated according to Eq. . As $p$ grows, coherent oscillations of a majority of spins begin to appear so that the magnetization $m$ shows noisy periodic or irregular oscillations. The noise is the manifestation of the stochasticity of the updating rule. Similar patterns can be seen in Ref. [@bagnoli2005], where the effect of the size of neighborhood is studied.

As shown in the following, by changing several parameters, we can obtain stochastic bifurcation diagrams similar to the mean-field ones. The following microscopic simulations were carried using lattices of $N=10,000$ sites, with a transient of $10,000$ time steps. The entropy $\eta$ was computed with 256 boxes and $25,600$ time steps.

In Fig. \[bifp\] we show the bifurcation map and the entropy $\eta$ as functions of $p$. There is always some disorder, even for small values of $p$ where $m\sim 0$, and as $p$ grows we find bifurcations and more disorder. Indeed, the entropy $\eta$ is a good measure of this behavior, small values of $\eta$ corresponding to noisy “periodic orbits” while larger ones to disorder (“chaos”).

In the mean-field case, we found 