  We derive bifurcation test equations for A-series singularities of nonlinear
functionals and, based on these equations, we propose a numerical method for
detecting high codimensional bifurcations in parameter-dependent PDEs such as
parameter-dependent semilinear Poisson equations. As an example, we consider a
Bratu-type problem and show how high codimensional bifurcations such as the
swallowtail bifurcation can be found numerically. In particular, our original
contributions are (1) the use of the Infinite-dimensional Splitting Lemma, (2)
the unified and simplified treatment of all A-series bifurcations, (3) the
presentation in Banach spaces, i.e. our results apply both to the PDE and its
(variational) discretization, (4) further simplifications for
parameter-dependent semilinear Poisson equations (both continuous and
discrete), and (5) the unified treatment of the continuous problem and its
discretisation.
