The two-site open Bose--Hubbard dimer model is a celebrated fundamental quantum optical model that accounts for the dynamics of bosons at two lossy interacting sites. Recently, two coupled, driven, and lossy photonic crystal nanocavities ---which are optical devices that operate with only a few hundred photons due to their extremely small size--- have been shown to realise this model experimentally. Thus, there is much interest in understanding the different behaviours that such model exhibits for theoretical and practical purposes.
This talk will show the different dynamics in the semiclassical approximation of this quantum optical system by presenting a comprehensive bifurcation analysis. We characterised different transitions of chaotic attractors in parameter plane by numerically computing tangency bifurcations between stable and unstable manifolds of saddle equilibria and periodic orbits. By doing so, we identify codimension-two degenerate singular cycles, and their generalisations, as responsible for the organisations of different tangency and heteroclinic bifurcations between saddle equilibria periodic orbits in parameter plane. Thus, we provide a roadmap for observable chaotic dynamics in the semiclassical approximation of the two-site Bose--Hubbard dimer model, which connects novel results in bifurcation theory with novel applications through numerical continuation techniques.