In this paper, we have investigated the dynamic behaviour of a non-adiabatic fed-batch reactor (FBR) in which an exothermic, irreversible first-order reaction is occurring. Fed-batch operation is a mode of operation in between that of a batch reactor and a CSTR. The transition towards each of these modes of operation as a function of the system parameters has been studied. The two parameters chosen for this purpose are the time of operation in the batch mode and the volume fraction which is added and removed at regular intervals of time. This system belongs to the class of systems with forcing occurring at discrete intervals of time. The bifurcation diagram has been generated using the shooting method in combination with a continuation method. The dynamics of the FBR are shown to be more complicated than the dynamics of the corresponding CSTR. We have observed different kinds of bifurcation in this system as the bifurcation parameter is varied. These include the classical period-doubling scenario, a period-adding sequence composed of mixed-mode oscillations giving rise to a region of alternating periodic and chaotic behaviour. Regions of quasi-periodic and intermittent behaviour also exist for some values of the bifurcation parameter. Crisis arising from the collision of a chaotic attractor with an unstable fixed point is shown to exist for this system.