$(-J,W)$ where $\eta>1/2$. ](lambda-J-W-PI00-o15-k20 "fig:"){width="0.9\columnwidth"}\
![\[fig:mfpdWJ\] (Color online) (a) Mean-field phase diagram of the Lyapunov exponent $\lambda$ showing the values of $(-J,W)$ where $\lambda>0$. (b) The mean-field phase diagram of the entropy showing the values of $(-J,W)$ where $\eta>1/2$. ](eta-J-W-PI00-o15-m8-k20 "fig:"){width="0.9\columnwidth"}

The dotted horizontal lines in Fig. \[fig:mf-J\] (c) and Fig. \[fig:mf-W\] (c) correspond to period 2, $\eta=1/8$, and period 4, $\eta=1/4$ orbits. Looking at the bifurcation diagram in Fig. \[fig:mf-J\] (a), for small $-J$, the map has a fixed point and as $-J$ grows there is a first bifurcation to a period 2 orbit and another one to what looks like a period 4 orbit, but $\eta=1/8$ instead of $\eta=1/4$ for period 4 orbits. What appears like a bifurcation to period four orbits is actually a pitchfork bifurcation to two period-two orbits that depend on the initial magnetization $m_0$ as mentioned before. There are other pitchfork bifurcations for other values of $J$ with $W$ fixed and also for values of $W$ with $J$ fixed.

Small-World stochastic bifurcations {#sec:simulations}
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In the Watts-Strogatz small-world model [@WattsStrogatz], starting with a network where every site has $K$ nearest neighbors, at any site $i$, with probability $p$, known as the long-range connection probability, 