Topological Phase Transitions in the Nonlinear Parallel Ising Model

We investigate some phase transitions of a nonlinear, parallel version of the Ising model, characterized by an antiferromagnetic linear coupling and a ferromagnetic nonlinear one. This model arises in problems of opin- ion formation. The mean-field approximation shows chaotic oscillations by changing the linear coupling or the connectivity. The spatial model ex- hibits bifurcations in the average magnetization, similar to what is seen in the mean-field approximation induced by the change of the topology, after rewiring short-range to long-range connections as predicted by the small-world effect. These coherent periodic and chaotic oscillations of the magnetization reflect a certain degree of synchronization of the spins, in- duced by long-range couplings. Similar bifurcations may be induced in the randomly connected model by changing the coupling or the connectivity and the synchronism of the updating (dilution of the rule).