Macroscopic dynamics and phase transitions in a bipartite mean-field model with averaged coupling

We investigate a dynamic bipartite mean-field spin system in which intra-group interactions are uniform while inter-group interactions are defined through their averaged strength. The model extends the classical Curie–Weiss framework and connects equilibrium results for bipartite systems with a dynamical Large Deviations (LDs) approach. Using Glauber single spin-flip dynamics, we establish a LD principle for the empirical trajectory measures and derive the corresponding McKean–Vlasov limit equations governing the macroscopic magnetizations in the thermodynamic limit. The resulting nonlinear system exhibits a rich bifurcation structure depending on the intra- and inter-group coupling parameters. We identify explicit critical thresholds separating regimes with one, three, five, seven, and nine stationary states. In the strongly coupled regime, we prove the existence of complex multistability and determine the stability of each equilibrium via spectral analysis of the linearized dynamics. The model reveals how averaged cross-interactions transfer polarization between subcritical and supercritical populations, leading to collective alignment. Our results provide a rigorous dynamical characterization of phase transitions in bipartite mean-field systems and offer a flexible framework for modeling interacting heterogeneous populations.