A method for studying the dynamics of periodically forced chemical reactors is presented along with the appropriate numerical techniques for its implementation. The method is capable of routinely treating many characteristics of forced oscillatory systems such as subharmonic and related bifurcations, bifurcations to invariant tori and frequency locking. It involves direct solution for the fixed points of the stroboscopic map. This procedure is particularly suitable for detecting certain routes to deterministic chaos such as the Feigenbaum period doubling cascade. Some illustrative examples from a homogeneous catalytic reactor model are included.