  Near a birth-death critical point in a one-parameter family of gradient
flows, there are precisely two Morse critical points of index difference one on
the birth side. This paper gives a self-contained proof of the folklore theorem
that these two critical points are joined by a unique gradient trajectory up to
time-shift. The proof is based on the Whitney normal form, a Conley index
construction, and an adiabatic limit analysis for an associated fast-slow
differential equation.
