The steady-state and dynamic behaviour of a single Pd---Al2O3 catalyst particle is studied for the selective hydrogenation of ethyne in the presence of ethene, without addition of carbon monoxide. The particle-to-gas heat transfer in the reactor is characterized. During selective hydrogenation, not only the ignition and the extinction phenomena but also oscillatory behaviour is observed. The nature of the single and multipeak oscillations is discussed. With a dynamic model, based on relatively simple kinetic equations and an additional slow mechanism, e.g. the formation of ethylidyne on the catalyst surface, the qualitative features of this system can be described.