This paper explores the influence of uncertainties and noise on the global dynamics of nonlinear systems, with emphasis on the basins and attractors’ topology and on the dynamic integrity of coexisting solutions. For this, the adaptive phase-space discretization strategy proposed in Part I  is employed. Two well-known archetypal oscillators found in engineering applications, the Helmholtz and Duffing oscillators under harmonic or parametric excitation with uncertain parameters or added load noise, are studied. The results demonstrate the adaptive capability of the proposed method, increasing the resolution without increasing the computational cost. Mean basins of attraction and attractors’ distributions are obtained. The time-dependency of stochastic responses is demonstrated, with long-transients influencing the short-term behavior. Finally, the effect of uncertainties and noise on the basins areas, attractors distributions and basin boundaries is discussed, which can be used to evaluate the dynamic integrity of the competing basins.