The parametric instability landscape of coupled Kerr parametric oscillators

We study networks of coupled Kerr parametric oscillators (KPOs) and map their stability regions using secular perturbation theory. Starting with two coupled KPOs, we show how bifurcations arise from the competition between parametric drive and linear coupling. We extend this to larger all-to-all coupled networks and find that bifurcation transitions become uniformly spaced in the thermodynamic limit. Our results delineate the precise bounds where KPO networks behave like Ising models, providing guidance for experimental implementation.