The tune shift with amplitude is an important concept in circular accelerators or synchrotrons.
The machine may be described via a symplectic one turn map at each position, which may be thought of as the Poincaire section of the dynamics.
A simple harmonic oscillator has a constant tune for all initial positions in phase space.
When the tune reaches resonant values, it can be unstable, and thus a tune-shift with amplitude can limit the stability region, or dynamic aperture.
In accelerator physics, both the transverse and the longitudinal dynamics show tune shift with amplitude.
A simple model of the transverse dynamics is of an oscillator with a single sextupole, it is referred to as the Hénon map.
It may also be computed by tracking the orbit through phase space, and then Fourier transforming the projections onto the different planes.