  Bose-Einstein condensates are described in a mean-field approach by the
nonlinear Gross-Pitaevskii equation and exhibit phenomena of nonlinear
dynamics. The eigenstates can undergo bifurcations in such a way that two or
more eigenvalues and the corresponding wave functions coalesce at critical
values of external parameters. E.g. in condensates without long-range
interactions a stable and an unstable state are created in a tangent
bifurcation when the scattering length of the contact interaction is varied. At
the critical point the coalescing states show the properties of an exceptional
point. In dipolar condensates fingerprints of a pitchfork bifurcation have been
discovered by Rau et al. [Phys. Rev. A, 81:031605(R), 2010]. We present a
method to uncover all states participating in a pitchfork bifurcation, and
investigate in detail the signatures of exceptional points related to
bifurcations in dipolar condensates. For the perturbation by two parameters,
viz. the scattering length and a parameter breaking the cylindrical symmetry of
the harmonic trap, two cases leading to different characteristic eigenvalue and
eigenvector patterns under cyclic variations of the parameters need to be
distinguished. The observed structures resemble those of three coalescing
eigenfunctions obtained by Demange and Graefe [J. Phys. A, 45:025303, 2012]
using perturbation theory for non-Hermitian operators in a linear model.
Furthermore, the splitting of the exceptional point under symmetry breaking in
either two or three branching singularities is examined. Characteristic
features are observed when one, two, or three exceptional points are encircled
simultaneously.
