By studying the effects of quadratic anisotropy and quartic perturbations on two-dimensional harmonic oscillators, 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.

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• Angular momentum of light
• Geometrical & wave optics
• Spatial profiles of optical beams
• Topological effects in photonic systems
• Bose-Hubbard model
• Perturbative methods
• Phase space methods
• Semiclassical methods