  In semiclassical studies of systems with mixed phase space, the neighbourhood
of bifurcations of co-dimension two is felt strongly even though such
bifurcations are ungeneric in classical mechanics. We discuss a scenario which
reveals this fact and derive the correct semiclassical contribution of the
bifurcating orbits to the trace of the unitary time evolution operator. That
contribution has a certain collective character rather than being additive in
the individual periodic orbits involved. The relevance of our observation is
demonstrated by a numerical study of the kicked top; the collective
contribution derived is found to considerably improve the semiclassical
approximation of the trace.
