Computer simulations of a detailed mechanism of the peroxidase-oxidase reaction show periodic-chaotic sequences for some ranges of enzyme concentrations. The mechanism involves two coupled feedback cycles that are both driven by oxygen. One of these cycles is the peroxidase catalytic cycle while the other is a conservation cycle involving all five known intermediates of the peroxidase enzyme. The reactions involving the enzyme intermediate ferroperoxidase are essential in generating complex and chaotic oscillations. This work demonstrates that the chaos previously observed in experiments can be accounted for by the known mechanism and kinetics of the reaction.