  Phase reduction is a well-established technique used to analyze the timing of
oscillations in response to weak external inputs. In the preceding decades, a
wide variety of results have been obtained for weakly perturbed oscillators
that place restrictive limits on the magnitude of the inputs or on the
magnitude of the time derivatives of the inputs. By contrast, no general
reduction techniques currently exist to analyze oscillatory dynamics in
response to arbitrary, large magnitude inputs and comparatively very little is
understood about these strongly perturbed limit cycle oscillators. In this
work, the theory of isostable reduction is leveraged to develop an adaptive
phase-amplitude transformation that does not place any restrictions on the
allowable input. Additionally, provided some of the Floquet multipliers of the
underlying periodic orbits are near-zero, the proposed method yields a
reduction in dimension comparable to that of other phase-amplitude reduction
frameworks. Numerical illustrations show that the proposed method accurately
reflects synchronization and entrainment of coupled oscillators in regimes
where a variety of other phase-amplitude reductions fail.
