Reduced Order Characterization of Nonlinear Oscillations Using an Adaptive Phase-Amplitude Coordinate Framework

TL;DR

This paper introduces a novel adaptive phase-amplitude coordinate framework for reduced order modeling of highly nonlinear oscillations, enabling accurate capture of complex dynamics beyond linear approximations.

Contribution

It presents a general strategy for reduced order modeling of nonlinear oscillations using phase-amplitude reduction, applicable to systems with arbitrary external inputs.

Findings

Outperforms standard linearization techniques in capturing nonlinear oscillations.

Accurately models systems with large amplitude oscillations.

Provides a unified approach for small and large oscillation regimes.

Abstract

We propose a general strategy for reduced order modeling of systems that display highly nonlinear oscillations. By considering a continuous family of forced periodic orbits defined in relation to a stable fixed point and subsequently leveraging phase-amplitude-based reduction strategies, we arrive at a low order model capable of accurately capturing nonlinear oscillations resulting from arbitrary external inputs. In the limit that oscillations are small, the system dynamics relax to those obtained from local linearization, i.e.,~that can be fully described using linear eigenmodes. For larger amplitude oscillations, the behavior can be understood in terms of the dynamics of a small number of nonlinear modes. We illustrate the proposed strategy in a variety of examples yielding results that are substantially better than those obtained using standard linearization-based techniques.