Sorting-free Hill-based stability analysis of periodic solutions through Koopman analysis

In this paper, we aim to study nonlinear time-periodic systems using the Koopman operator, which provides a way to approximate the dynamics of a nonlinear system by a linear time-invariant system of higher order. We propose for the considered system class a specific choice of Koopman basis functions combining the Taylor and Fourier bases. This basis allows to recover all equations necessary to perform the harmonic balance method as well as the Hill analysis directly from the linear lifted dynamics. The key idea of this paper is using this lifted dynamics to formulate a new method to obtain stability information from the Hill matrix. The error-prone and computationally intense task known by sorting, which means identifying the best subset of approximate Floquet exponents from all available candidates, is circumvented in the proposed method. The Mathieu equation and an n-DOF generalization are used to exemplify these findings.

Automated summary

This paper aims to study nonlinear time-periodic systems using the Koopman operator, which approximates the dynamics of a nonlinear system by a higher-order linear time-invariant system. A specific choice of Koopman basis functions, incorporating the Taylor and Fourier bases, is proposed for the considered system class. This basis allows recovery of all necessary equations for harmonic balance method and Hill analysis directly from the linear lifted dynamics. The key idea of this paper is to use this lifted dynamics to formulate a new method to obtain stability information from the Hill matrix, circumventing the error-prone and computationally intense task of sorting.