Tectonic faults typically break in a single rupture mode within the range of styles from slow slip to dynamic earthquake failure. However, in increasingly well-documented instances, the same fault segment fails in both slow and fast modes within a short period, as in the sequences that culminated in the 2011 Mw = 9.0 Tohoku-Oki, Japan, and 2014 Mw = 8.2 Iquique, Chile, earthquakes. Why slow slip alternates with dynamic rupture in certain regions but not in others is not well understood. Here, we integrate laboratory experiments and numerical simulations to investigate the physical conditions leading to cycles where the two rupture styles alternate. We show that a bifurcation takes place near the stability transition with sequences encompassing various rupture modes under constant loading rate. The range of frictional instabilities and slip cycles identified in this study represents important end-members to understand the interaction of slow and fast slip on the same fault segment.