A global study of the occurrence of steady state multiplicity and limit cycles in gas phase polyethylene reactors is carried out using elementary concepts of singularity theory and continuation techniques, for both open loop and closed loop operations. The well-mixed model of the unit consisting of the reactor, the heat exchanger and the ethylene partial-pressure controller, is reduced analytically to one single algebraic equation. The singularity theory is efficiently used for the classification, in multidimensional parameter space, of the different static and dynamic behaviour exhibited by the unit. This includes regions of unique and stable behaviour, regions of hysteresis, regions of limit cycles as well as regions of bistability resulting from interactions between hysteresis and limit cycles. A useful and practical picture of the effect of the different kinetic and operating parameters of the unit is constructed. The tools developed for a single site type are also extended to study the dynamics induced by multiple catalytic site types. The dynamics of the process under a conventional temperature PI controller are also studied. It is shown that while the integral action eliminates the steady state multiplicity, it introduces instability for some ranges of catalyst feed rates. The stability analysis allows the construction of practical diagrams showing the effect of model parameters on the closed loop stability behaviour of the process.