The cell division cycle requires oscillations in activity of B-type cyclin (Clb)-Cdk1 kinases. Oscillations are due to periodic cyclin degradation by the anaphase-promoting complex (APC) activated by Cdc20 or Cdh1, and to cyclical accumulation of the Sic1 inhibitor. The results presented here suggest that the regulatory machinery controlling Clb kinase levels embeds two distinct oscillatory mechanisms. One, a "relaxation oscillator," involves alternation between two meta-stable states: Clb high/inhibitors (Sic1/APC-Cdh1) low, and Clb low/inhibitors high. The other, a "negative feedback oscillator," involves Clb kinase activation of APC-Cdc20, leading to Clb degradation. Genetic analysis suggests that these two mechanisms can function independently, and inactivation of both mechanisms is required to prevent mitosis. Computational modeling confirms that two such mechanisms can be linked to yield a robust cell cycle control system.