  We examine the dynamics of a dissipative photonic Josephson junction formed
by the weak coupling of an optical soliton in a nonlinear dielectric waveguide
and a co-propagating surface plasmon along a parallel metal surface with a
linear dielectric spacer. We employ a heuristic model with a coupling function
that depends on the soliton amplitude, and consider two phenomenological
dissipation mechanisms separately: angular velocity dissipation and population
imbalance dissipation. In the former dissipation mechanism, the system exhibits
phase-slip phenomenon where the odd-\pi phase modes decay into even-\pi phase
modes. The latter damping mechanism sculptures the phase-space significantly by
introducing complex features, among which Hopf type bifurcations are notable.
We show that some of the bifurcation points expand to stable limit cycles for
certain regimes of the model parameters.
