A spectral characterization of nonlinear normal modes

G.I. Cirillo, A. Mauroy, L. Renson, G. Kerschen, R. Sepulchre

Research output: Contribution to journalArticle (Academic Journal)peer-review

33 Citations (Scopus)
299 Downloads (Pure)


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.
Original languageEnglish
Pages (from-to)284-301
Number of pages18
JournalJournal of Sound and Vibration
Early online date24 May 2016
Publication statusPublished - 1 Sept 2016


  • Nonlinear normal modes
  • Koopman operator
  • Spectral
  • Invariant manifolds
  • Parametrization


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