  We perform a systematic study of the temporal dynamics emerging in the
asymmetrically driven dissipative Bose-Hubbard dimer model. This model
successfully describes the nonlinear dynamics of photonic diatomic molecules in
linearly coupled Kerr resonators coherently excited by a single laser beam.
Such temporal dynamics include self-pulsing oscillations, period doubled
oscillatory states, chaotic dynamics, and spikes. The different states and
dynamical regimes have been thoroughly characterized using bifurcation
analysis. This analysis has allowed us to identify the main instabilities, i.e.
bifurcations, responsible for the appearance of the previously stated dynamics.
