  We investigate the structure of resonance widths of a Bose-Hubbard Dimer with
intersite hopping amplitude $k$, which is coupled to continuum at one of the
sites with strength $\gamma$. Using an effective non-Hermitian Hamiltonian
formalism, we show that by varying the on-site interaction term $\chi$ the
resonances undergo consequent bifurcations. For $\Lambda=k/\gamma\geq 0.5$, the
bifurcation points follow a scaling law ${\tilde \chi}_n \equiv \chi_n N/k =
f_{\Lambda}(n-0.5/\Lambda)$, where $N$ is the number of bosons. For the
function $f_{\Lambda}$ two different $\Lambda$ dependences are found around the
minimum and the maximum bifurcation point.
