Discontinuity-induced bifurcation cascades in flows and maps with application to models of the yeast cell cycle

• We investigate bifurcation cascades in models of the eukaryotic cell cycle.
• We develop a general theory for classes of period-adding cascades in piecewise-defined maps with gaps.
• The theory predicts characteristic scalings of bifurcation values that agree with numerical observations.
• We uncover global saddle–node bifurcations in piecewise-defined maps and piecewise-smooth hybrid dynamical systems.

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.