Nonlinear normal modes and spectral submanifolds: Existence, uniqueness and use in model reduction

TL;DR

This paper develops a comprehensive mathematical framework for nonlinear modal analysis in dissipative systems, establishing existence, uniqueness, and robustness of nonlinear normal modes and spectral submanifolds for model reduction.

Contribution

It introduces a unified approach that covers autonomous and time-dependent systems, clarifies the definitions of NNMs and SSMs, and provides rigorous existence and uniqueness results based on spectral quotients.

Findings

Existence and uniqueness of SSMs depend on spectral quotients.

Damping is crucial for the existence and robustness of NNMs and SSMs.

The framework applies to both autonomous and time-dependent dissipative oscillatory systems.

Abstract

We propose a unified approach to nonlinear modal analysis in dissipative oscillatory systems. This approach eliminates conflicting definitions, covers both autonomous and time-dependent systems, and provides exact mathematical existence, uniqueness and robustness results. In this setting, a nonlinear normal mode (NNM) is a set filled with small-amplitude recurrent motions: a fixed point, a periodic orbit or the closure of a quasiperiodic orbit. In contrast, a spectral submanifold (SSM) is an invariant manifold asymptotic to a NNM, serving as the smoothest nonlinear continuation of a spectral subspace of the linearized system along the NNM. The existence and uniqueness of SSMs turns out to depend on a spectral quotient computed from the real part of the spectrum of the linearized system. This quotient may well be large even for small dissipation, thus the inclusion of damping is essential for firm conclusions about NNMs, SSMs and the reduced-order models they yield.