  The first order optimality conditions of optimal control problems (OCPs) can
be regarded as boundary value problems for Hamiltonian systems. Variational or
symplectic discretisation methods are classically known for their excellent
long term behaviour. As boundary value problems are posed on intervals of
fixed, moderate length, it is not immediately clear whether methods can profit
from structure preservation in this context. When parameters are present,
solutions can undergo bifurcations, for instance, two solutions can merge and
annihilate one another as parameters are varied. We will show that generic
bifurcations of an OCP are preserved under discretisation when the OCP is
either directly discretised to a discrete OCP (direct method) or translated
into a Hamiltonian boundary value problem using first order necessary
conditions of optimality which is then solved using a symplectic integrator
(indirect method). Moreover, certain bifurcations break when a non-symplectic
scheme is used. The general phenomenon is illustrated on the example of a cut
locus of an ellipsoid.
