An Adaptive Phase-Amplitude Reduction Framework Without $\mathcal{O}(\epsilon)$ Constraints on Inputs

TL;DR

This paper introduces an adaptive phase-amplitude reduction method for oscillators that works with large, arbitrary inputs, overcoming previous limitations on input size and derivative constraints, and accurately predicts synchronization.

Contribution

It develops a novel reduction framework leveraging isostable reduction that removes input restrictions and maintains accuracy for strongly perturbed oscillators.

Findings

Accurately predicts synchronization and entrainment in strongly perturbed oscillators.

Works without restrictions on input magnitude or derivatives.

Maintains dimension reduction comparable to existing methods.

Abstract

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.