the dilution $d$ with $J=-10$, and different values of the long-range connection probability $p$.](PI5-o4-N10000-k20-J-10p00-W15p00-p0p50 "fig:"){width="0.45\columnwidth"}   ![\[bifd\] (Color online). Bifurcation diagrams of the magnetization $m$ on small-world networks, dots (in magenta), and the entropy $\eta$, continuous curve (in green), as functions of the dilution $d$ with $J=-10$, and different values of the long-range connection probability $p$.](PI5-o4-N10000-k20-J-10p00-W15p00-p1p00 "fig:"){width="0.45\columnwidth"}
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Heterogeneity
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In order to measure the effects of heterogeneity, we let a fraction $\xi$ of spins interact ferromagnetically ($J>0$) with their $K$ neighbors and a fraction $1-\xi$ interact antiferromagnetically ($J<0$). We show in Fig. \[fig:mix\] the bifurcation diagrams of the magnetization $m$ together with the entropy $\eta$ as functions of $\xi$ for different values of $p$. Again, the entropy is a good measure of disorder. One can see a “bubbling” effect very similar to what observed by changing the dilution. In other words, oscillations, which are a product of antiferromagnetism and parallelism, are actually favoured by a small percentage of asynchronism and/or of ferromagnetic nodes, for a partial long-range rewiring of links.

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                                                                                                                                                                                                                                                           $p=0.05$                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     $p=0.2$
   ![\[fig:mix\] (Color online.) Small-world ferro-anti ferro bifurcation diagram (left axis, dots in magenta) and entropy $\eta$ (right axis, continuous curve in green) of the magnetization 