  Saffman and Turner (1957) argued that the collision rate for droplets in
turbulence increases as the turbulent strain rate increases. But the numerical
simulations of Dhanasekaran et al. (2021) in a steady straining flow show that
the Saffman-Turner model is oversimplified because it neglects droplet-droplet
interactions. These result in a complex dependence of the collision rate on the
strain rate and on the differential settling speed. Here we show that this
dependence is explained by a sequence of bifurcations in the collision
dynamics. We compute the bifurcation diagram when strain is aligned with
gravity, and show that it yields important insights into the collision
dynamics. First, the steady-state collision rate remains non-zero in the limit
Kn $\to0$, contrary to the common assumption that the collision rate tends to
zero in this limit (Kn is a non-dimensional measure of the mean free path of
air). Second, the non-monotonic dependence of the collision rate on the
differential settling speed is explained by a grazing bifurcation. Third, the
bifurcation analysis explains why so-called "closed trajectories" appear and
disappear. Fourth, our analysis predicts strong spatial clustering near certain
saddle points, where the effects of strain and differential settling cancel
