The stability of parametrically forced coupled oscillators in sum resonance

For a system of two damped parametrically forced oscillators in sum resonance the planar stability diagram of amplitude versus frequency of the forcing shows a discontinuity at damping zero. This is a well known phenomenon, for which we give a geometrical explanation. A linear stability analysis suffices. We show that a versal (i.e. a structurally stable) matrix unfolding for this problem needs four parameters, indicating that the stability diagram is actually four dimensional. The boundary of the stability region in parameter space is singular, this provides a geometric explanation of the discontinuity in the planar stability diagram.