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This paper applies methods of numerical continuation analysis to document characteristic bifurcation cascades of limit cycles in piecewise-smooth, hybrid-dynamical-system models of the eukaryotic cell cycle, and associated period-adding cascades in piecewise-defined maps with gaps. A general theory is formulated for the occurrence of such cascades, for example given the existence of a period-two orbit with one point on the system discontinuity and with appropriate constraints on the forward trajectory for nearby initial conditions. In this case, it is found that the bifurcation cascade for nearby parameter values exhibits a scaling relationship governed by the largest-in-magnitude Floquet multiplier, here required to be positive and real, in complete agreement with the characteristic scaling observed in the numerical study. A similar cascade is predicted and observed in the case of a saddle-node bifurcation of a period-two orbit, away from the discontinuity, provided that the associated center manifold is found to intersect the discontinuity transversally.