for different values of the long range probability $p$ and $N=10,000$, $|J|=10$,. for $P=0.05$ there is no threshold for $\eta$, for $p=0.2$ it is $\eta=0.7$, for $p=0.8$ and $p=1.0$ it is $\eta=0.6$.](sw-bif-eta-m-xi-PI5-o8-N10000-k20-J-10p00-W15p00-p1p00 "fig:"){width="0.45\columnwidth"}
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Conclusions {#sec:conclusions}
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We investigated the phase transitions of a nonlinear, parallel version of the Ising model, characterized by a linear coupling $J<0$ and a nonlinear one $W>0$. The mean-field approximation shows chaotic oscillations, by changing the couplings $J$ and $W$ or the connectivity $K$. We showed in the Appendix that there is a scaling relation among these parameters.

The nonlinear Ising model was studied on small-world networks, where $p$ is the probability of long-range rewiring of links. Here, entropy of the magnetization becomes a measure of disorder which is adequate once a threshold between the presence and absence of noisy periodic orbits is established. The noisy periodic and disordered behavior of $m$ imply a certain degree of synchronization of the spins, induced by long-range couplings.

We have shown also that similar bifurcations may be induced in the randomly connected model by changing the parameters $J$, the dilution factor $d$ and the heterogeneity $\xi$, by mixing ferromagnetic and antiferromagnetic interactions.

In particular, we observed that 