Bifurcations of the conjugate locus

The conjugate locus of a point p in a surface 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.