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

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## 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.

## Keywords

Nonlinear normal modes Invariant manifolds Model reduction## Notes

### Acknowledgments

We are grateful to Rafael de la Llave and Alex Haro for detailed technical explanations on their invariant manifold results, to Ludovic Renson for clarifying the numerical approach in Ref. [36] and to Paolo Tiso for helpful discussions on nonlinear normal modes. We are also thankful to Alireza Hadjighasem for his advice on visualization and to Robert Szalai for pointing out typographical errors in an earlier version of this manuscript. Finally, we are pleased to acknowledge useful suggestions from the two anonymous reviewers of this work.

## Supplementary material

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