  We show that symplectic integrators preserve bifurcations of Hamiltonian
boundary value problems and that nonsymplectic integrators do not. We provide a
universal description of the breaking of umbilic bifurcations by nonysmplectic
integrators. We discover extra structure induced from certain types of boundary
value problems, including classical Dirichlet problems, that is useful to
locate bifurcations. Geodesics connecting two points are an example of a
Hamiltonian boundary value problem, and we introduce the jet-RATTLE method, a
symplectic integrator that easily computes geodesics and their bifurcations.
Finally, we study the periodic pitchfork bifurcation, a codimension-1
bifurcation arising in integrable Hamiltonian systems. It is not preserved by
either symplectic on nonsymplectic integrators, but in some circumstances
symplecticity greatly reduces the error.
