“bubbling” transition: the oscillations are favored, for intermediate values of the rewiring $p$, by a non-complete parallelism.

As shown in the figure, for values of $p$ larger than $0.1$, the dilution is able to trigger bifurcations also in the spatial model. In contrast with the linear Ising model [@Cirillo], where even a small amount of asynchronism is able to destroy the “effective” antiferromagnetic coupling, here the behavior is smooth with respect to dilution. See Ref. [@SS-meta] for a study about metastable effects in the linear model.

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                                                                                                                                                                                  $p=0.1$                                                                                                                                                                                                                                                                                                                                                                 $p=0.2$
   ![\[bifd\] (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-p0p10 "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-p0p20 "fig:"){width="0.45\columnwidth"}
                                                                                                                                                                                  $p=0.5$                                                                                                                                                                                                                                                                                                                                                                 $p=1.0$
   ![\[bifd\] (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 