Novak and Tyson have proposed a realistic mathematical model of the biochemical mechanism that regulates M-phase promoting factor (MPF), the major enzymatic activity controlling mitotic cycles in frog eggs, early embryos, and cell-free egg extracts. We use bifurcation theory and numerical methods (AUTO) to characterize the codimension-one and -two bifurcation sets in this model. Our primary bifurcation parameter is the rate constant for cyclin synthesis, which can be manipulated experimentally by adding exogenously synthesized cyclin mRNA to extracts depleted of all endogenous mRNA molecules. For the secondary bifurcation parameter we use the total amount of one of the principal regulatory enzymes in the extract (APC, the enzyme complex that labels cyclin for degradation; Wee1, the kinase that inhibits MPF; or Cdc25, the phosphatase that activates MPF). We find a rich array of physiologically distinct behaviors exhibited by the model as these parameters are varied around values that seem plausible for frog eggs and extracts. In addition to unique, stable steady states (cell cycle arrest) and limit cycle oscillations (autonomous, periodic cell division), we find parameter combinations where the control system is bistable. For instance, an interphase-arrested state may coexist with a metaphase-arrested state, or two stable limit cycles of different amplitude and period may coexist. We suggest that such strange behavior is nearly unavoidable in a complex regulatory system like the cell cycle. Perhaps cells exploit some of these exotic bifurcations for control purposes that are as yet unrecognized by physiologists.