  Fast-slow systems with three slow variables and gradient structure in the
fast variables have, generically, hyperbolic umbilic, elliptic umbilic or
swallowtail singularities. In this article we provide a detailed local analysis
of a fast-slow system near a hyperbolic umbilic singularity. In particular, we
show that under some appropriate non-degeneracy conditions on the slow flow,
the attracting slow manifolds jump onto the fast regime and fan out as they
cross the hyperbolic umbilic singularity. The analysis is based on the blow-up
technique, in which the hyperbolic umbilic point is blown up to a 5-dimensional
sphere. Moreover, the reduced slow flow is also blown up and embedded into the
blown-up fast formulation. Further, we describe how our analysis is related to
classical theories such as catastrophe theory and constrained differential
equations.
