Stochastic Bifurcations in the Nonlinear Parallel Ising Model

TL;DR

This paper explores phase transitions and bifurcations in a nonlinear parallel Ising model with mixed couplings, revealing complex oscillatory behaviors influenced by network topology, coupling types, and inhomogeneity, relevant to opinion dynamics.

Contribution

It introduces a nonlinear parallel Ising model with mixed couplings and analyzes its bifurcation behavior under various network topologies and inhomogeneities, highlighting new dynamical phenomena.

Findings

Chaotic oscillations in mean-field approximation.

Bifurcations induced by changing network topology.

Bubbling bifurcation behavior with inhomogeneity.

Abstract

We investigate the phase transitions of a nonlinear, parallel version of the Ising model, characterized by an antiferromagnetic linear coupling and ferromagnetic nonlinear one. This model arises in problems of opinion formation. The mean-field approximation shows chaotic oscillations, by changing the couplings or the connectivity. The spatial model shows bifurcations in the average magnetization, similar to what seen in the mean-field approximation, induced by the change of the topology, after rewiring short-range to long-range connection, 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, induced by long-range couplings. Similar bifurcations may be induced in the randomly connected model by changing the couplings or the connectivity and also the dilution (degree of asynchronism) of the updating. We also examined the effects of inhomogeneity, mixing ferromagnetic and antiferromagnetic coupling, which induces an unexpected bifurcation diagram with a "bubbling" behaviour, as also happens for dilution.