modes, presented in Fig. \[bifurcation diagrams\], were verified, in addition to the computation of eigenvalues for small perturbations, by direct simulations, performed by dint of the finite-difference algorithm. The instability development of unstable symmetric states was catalyzed by adding small initial symmetry-breaking perturbations to them. This was done for the unstable symmetric states with both $k>0$ and for $k<0$, i.e., self-trapped and leaky ones. Typical examples are displayed in Fig. \[unstable evolution\], in which the unstable symmetric modes feature single- and double (split)-peak shapes at small and large values of the inner-barrier’s height, $A=0.05$ and $A=0.5$, respectively (in the top and bottom rows of the figure).

As mentioned above, the instability of symmetric states is accounted for by pure imaginary eigenfrequencies, hence the originally developing instability is not oscillatory. However, the nonlinearity makes the unstable dynamics oscillatory, as seen in Fig. \[unstable evolution\]. In other words, the unstable symmetric modes spontaneously develop bosonic Josephson oscillations. Close to the instability onset, the effective oscillation period is very large (as it diverges precisely at the onset point), gradually decreasing deeper into the instability region. The dynamical symmetry breaking induced by the weak and moderate instability is incomplete, leading to periodic oscillations between 