  We consider a two-component linearly-coupled system with the intrinsic cubic
nonlinearity and the harmonic-oscillator (HO) confining potential. The system
models binary settings in BEC and optics. In the symmetric system, with the HO
trap acting in both components, we consider Josephson oscillations (JO)
initiated by an input in the form of the HO's ground state (GS) or dipole mode
(DM), placed in one component. With the increase of the strength of the
self-focusing nonlinearity, spontaneous symmetry breaking (SSB) between the
components takes place in the dynamical JO state. Under still stronger
nonlinearity, the regular JO initiated by the GS input carry over into a
chaotic dynamical state. For the DM input, the chaotization happens at smaller
powers than for the GS, which is followed by SSB at a slightly stronger
nonlinearity. In the system with the defocusing nonlinearity, SSB does not take
place, and dynamical chaos occurs in a small area of the parameter space. In
the asymmetric half-trapped system, with the HO potential applied to a single
component, we first focus on the spectrum of confined binary modes in the
linearized system. The spectrum is found analytically in the limits of weak and
strong inter-component coupling, and numerically in the general case. Under the
action of the coupling, the existence region of the confined modes shrinks for
GSs and expands for DMs. In the full nonlinear system, the existence region for
confined modes is identified in the numerical form. They are constructed too by
means of the Thomas-Fermi approximation, in the case of the defocusing
nonlinearity. Lastly, particular (non-generic) exact analytical solutions for
confined modes, including vortices, in one- and two-dimensional asymmetric
linearized systems are found. They represent bound states in the continuum.
