  By studying the effects of quadratic anisotropy and quartic perturbations on
the two-dimensional harmonic oscillator, one arrives at a simple model termed
here the Ince oscillator, whose analytic solutions are given in terms of Ince
polynomials. This one model unifies diverse physical systems, including
aberrated optical cavities that are shown to support Ince-Gauss beams as their
modes, and the two-mode Bose-Hubbard dimer describing two coupled superfluids.
The Ince oscillator model describes a topological transition which can have
very different origins: in the optical case, which is fundamentally linear, it
is driven by the ratio of astigmatic to spherical mirror aberrations, whereas
in the superfluid case it is driven by the ratio of particle tunneling to
interparticle interactions and corresponds to macroscopic quantum self
trapping.
