We solve the equations that describe, in a simplified way, the Newtonian dynamics of a self-gravitating non-rotating spheroidal body after loss of stability. We find that contraction to a singularity occurs only in a pure spherical collapse, and deviations from spherical symmetry stop the contraction through the stabilizing action of non-linear non-spherical oscillations. A real collapse occurs after damping of the oscillations because of energy losses, shock wave formation or viscosity. Detailed analysis of the non-linear oscillations is performed using a Poincaré map construction. Regions of regular and chaotic oscillations are localized on this map.