One- and two-parameter Hopf bifurcation behavior is analyzed for several variants of the Citri–Epstein mechanism of the chlorite–iodide reaction. The coefficients of an equation for the amplitude of oscillations (the universal unfolding of the Hopf bifurcation) are evaluated numerically. Local bifurcation diagrams and bifurcation sets are derived from the amplitude equation. Sub- and supercritical Hopf bifurcations are identified for the general case of a nondegenerate (codimension one) bifurcation. At degenerate (codimension two) points, the necessary higher-order terms are included in the unfolding, and features such as locally isolated branches of periodic orbits and bistability of a periodic orbit and a steady state are found. Inferences about the global periodic orbit structure and the existence of nearby codimension three Hopf bifurcation points are drawn by piecing together the local information contained in the unfoldings. Hypotheses regarding the global periodic orbit structure are tested using continuation methods to compute entire branches of orbits. A thorough discussion of the application of these methods is presented for one version of the mechanism, followed by a comparison of the complete two-parameter steady state bifurcation structure of three versions of the mechanism. In all cases, the potential for experimental verification of the predicted dynamics is examined.