In the present paper, we study by direct numerical simulation (DNS) and theoretical analysis, the dynamics of a fountain penetrating a pycnocline (a sharp density interface) in a density-stratified fluid. A circular, laminar jet flow of neutral buoyancy is considered, which propagates vertically upwards towards the pycnocline level, penetrates a distance into the layer of lighter fluid, and further stagnates and flows down under gravity around the up-flowing core thus creating a fountain. The DNS results show that if the Froude number (Fr) is small enough, the fountain top remains axisymmetric and steady. However, if Fr is increased, the fountain top becomes unsteady and oscillates in a circular flapping (CF) mode, whereby it retains its shape and moves periodically around the jet central axis. If Fr is increased further, the fountain top rises and collapses chaotically in a bobbing oscillation mode (or B-mode). The development of these two modes is accompanied by the generation of different patterns of internal waves (IW) in the pycnocline. The CF-mode generates spiral internal waves, whereas the B-mode generates IW packets with a complex spatial distribution. The dependence of the amplitude of the fountain-top oscillations on Fr is well described by a Landau-type two-mode-competition model.