We provide a complete study of the phase diagram characterising the distinct dynamical regimes emerging in a three-dimensional Josephson junction in an ultracold quantum gas. Considering trapped ultracold superfluids separated into two reservoirs by a barrier of variable height and width, we analyse the population imbalance dynamics following a variable initial population mismatch. We demonstrate that as the chemical potential difference is increased, the system transitions from Josephson plasma oscillations to either a dissipative (in the limit of low and narrow barriers) or a self-trapped regime (for large and wider barriers), with a crossover between the dissipative and the self-trapping regimes which we explore and characterize for the first time. This work, which extends beyond the validity of the standard two-mode model, connects the role of the barrier width, vortex rings and associated acoustic emission with different regimes of the superfluid dynamics across the junction, establishing a framework for its experimental observation, which is found to be within current experimental reach.
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