A spectral characterization of nonlinear normal modes

TL;DR

This paper links nonlinear normal modes to Koopman operator eigenfunctions, providing a new global parametrization that remains valid during manifold folding, thus enabling analysis at higher energy levels.

Contribution

It introduces a novel spectral approach connecting NNMs with Koopman eigenfunctions, allowing for a global parametrization valid during manifold folding.

Findings

Eigenfunctions of the Koopman operator correspond to NNMs.

The new parametrization remains valid during manifold folding.

Application demonstrated on a two-degree-of-freedom cubic nonlinear system.

Abstract

This paper explores the relationship that exists between nonlinear normal modes (NNMs) defined as invariant manifolds in phase space and the spectral expansion of the Koopman operator. Specifically, we demonstrate that NNMs correspond to zero level sets of specific eigenfunctions of the Koopman operator. Thanks to this direct connection, a new, global parametrization of the invariant manifolds is established. Unlike the classical parametrization using a pair of state-space variables, this parametrization remains valid whenever the invariant manifold undergoes folding, which extends the computation of NNMs to regimes of greater energy. The proposed ideas are illustrated using a two-degree-of-freedom system with cubic nonlinearity.