  The conjugate locus of a point $p$ in a surface $\mathcal{S}$ will have a
certain number of cusps. As the point $p$ is moved in the surface the conjugate
locus may spontaneously gain or lose cusps. In this paper we explain this
`bifurcation' in terms of the vanishing of higher derivatives of the
exponential map; we derive simple equations for these higher derivatives in
terms of scalar invariants; we classify the bifurcations of cusps in terms of
the local structure of the conjugate locus; and we describe an intuitive
picture of the bifurcation as the intersection between certain contours in the
tangent plane.
