Nonlinear Dynamics

, Volume 86, Issue 3, pp 1493–1534 | Cite as

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

  • George Haller
  • Sten Ponsioen
Original Paper


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.


Nonlinear normal modes Invariant manifolds Model reduction 



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

11071_2016_2974_MOESM1_ESM.gif (9.3 mb)
Supplementary material 1 (gif 9566 KB)
11071_2016_2974_MOESM2_ESM.gif (5.6 mb)
Supplementary material 2 (gif 5685 KB)


  1. 1.
    Arnold, V.I.: Geometrical Methods in the Theory of Ordinary Differential Equations. Springer, New York (1988)CrossRefGoogle Scholar
  2. 2.
    Arnold, V.I.: Mathematical Methods of Classical Mechanics. Springer, New York (1989)CrossRefGoogle Scholar
  3. 3.
    Avramov, K.V., Mikhlin, Y.V.: Nonlinear normal modes for vibrating mechanical systems. Review of theoretical developments. ASME Appl. Mech. Rev. 65, 060802-1 (2010)Google Scholar
  4. 4.
    Avramov, K.V., Mikhlin, Y.V.: Review of applications of nonlinear normal modes for vibrating mechanical systems ASME. Appl. Mech. Rev. 65, 020801-1 (2013)CrossRefGoogle Scholar
  5. 5.
    Belaga, E.G.: On the reducibility of a system of differential equations in the neighborhood of a quasiperiodic motion. Sov. Math. Dokl. 143(2), 255–258 (1962)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Blanc, F., Touze, C., Mercier, J.F., Ege, K., Bonnet Ben-Dhia, A.S.: On the numerical computation of nonlinear normal modes for reduced-order modelling of conservative vibratory systems. Mech. Syst. Signal Process. 36, 520–539 (2013)CrossRefGoogle Scholar
  7. 7.
    Gabale, A.P., Sinha, S.C.: Model reduction of nonlinear systems with external periodic excitations via construction of invariant manifolds. J. Sound Vib. 330, 2596–2607 (2011)CrossRefGoogle Scholar
  8. 8.
    Cabré, P., Fontich, E., de la Llave, R.: The parametrization method for invariant manifolds I: manifolds associated to non-resonant spectral subspaces. Indiana Univ. Mathe. J. 52, 283–328 (2003)CrossRefzbMATHGoogle Scholar
  9. 9.
    Cabré, P., Fontich, E., de la Llave, R.: The parameterization method for invariant manifolds III: overview and applications. J. Differ. Equ. 218, 444–515 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Cirillo, G.I., Mauroy, A., Renson, L., Kerschen, G., Sepulchre, R.: Global parametrization of the invariant manifold defining nonlinear normal modes using the Koopman operator, In: Proceedings ASME 2015 International Design Engineering Technical Conferences and Computers and Information in Engineering, Boston (2015)Google Scholar
  11. 11.
    Cirillo, G.I., Mauroy, A., Renson, L., Kerschen, G., Sepulchre, R.: A spectral characterization of nonlinear normal modes. J. Sound Vib. 377, 284–301 (2016)CrossRefGoogle Scholar
  12. 12.
    Elmegard, M.: Mathematical Modeling and Dimension Reduction in Dynamical Systems. Ph.D. thesis, Technical University of Denmark (2014)Google Scholar
  13. 13.
    Euler, L.: De seriebus divergenti bus, Opera omnia, Ser. 1,14, vol. 247, pp. 585–617. Leipzig, Berlin (1924)Google Scholar
  14. 14.
    Evans, L.C.: Partial Differ. Equ. AMS Press, Rhode Island (1998)Google Scholar
  15. 15.
    Fenichel, N.: Persistence and smoothness of invariant manifolds for flows. Indiana Univ. Math. J. 21, 193–226 (1971)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Guckenheimer, J., Holmes, P.: Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer, New York (1983)CrossRefzbMATHGoogle Scholar
  17. 17.
    Lyapunov, A.M.: The general problem of the stability of motion. Int. J. Control 55, 531–534 (1992)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Haro, A., de la Llave, R.: \({\cal A}\) parameterization method for the computation of invariant tori and their whiskers in quasi-periodic maps: Rigorous results. Differ. Equ. 228, 530–579 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Haro, A., Canadell, M., Figueras, J.-L., Luque, A., Mondelo, J.M.: The Parameterization Method for Invariant Manifolds: From Rigorous Results to Effective Computations. Springer, New York (2016)CrossRefzbMATHGoogle Scholar
  20. 20.
    Hirsch, M.W., Pugh, C.C., Shub, M.: Invariant Manifolds. Lecture Notes Math., vol. 583. Springer, New York (1977)Google Scholar
  21. 21.
    Hirsch, W., Smale, S., Devaney, R.L.: Differential Equations, Dynamical Systems, and an Introduction to Chaos, 3rd edn. Academic Press, Oxford (2013)zbMATHGoogle Scholar
  22. 22.
    Jiang, D., Pierre, C., Shaw, S.W.: Nonlinear normal modes for vibratory systems under harmonic excitation. J. Sound Vib. 288, 791–812 (2005)CrossRefGoogle Scholar
  23. 23.
    Kelley, A.F.: Analytic two-dimensional subcenter manifolds for systems with an integral. Pac. J. Math. 29, 335–350 (1969)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Kerschen, G., Peeters, M., Golinval, J.C., Vakakis, A.F.: Nonlinear normal modes. Part I: a useful framework for the structural dynamicist. Mech. Syst. Signal Process. 23, 170–194 (2009)CrossRefGoogle Scholar
  25. 25.
    Kerschen, G. (ed.): Modal Analysis of Nonlinear Mechanical Systems. Springer, Berlin (2014)zbMATHGoogle Scholar
  26. 26.
    Kuether, R.J., Renson, L., Detroux, T., Grappasonni, C., Kerschen, G., Allen, M.S.: Nonlinear normal modes, modal interactions and isolated resonance curves. J. Sound Vib. 351, 299–310 (2015)CrossRefGoogle Scholar
  27. 27.
    Mezić, I.: Spectral properties of dynamical systems, model reduction and decompositions. Nonlinear Dyn. 41, 309–325 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Mireles-James, J.D.: Polynomial approximation of one parameter families of (un)stable manifolds with rigorous computer assisted error bounds. Indag. Math. 26, 225–265 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Nayfeh, A.H.: Perturbation Methods. Wiley, New York (2004)Google Scholar
  30. 30.
    Neild, S.A., Champneys, A.R., Wagg, D.J., Hill, T.L., Cammarano, A.: The use of normal forms for analysing nonlinear mechanical vibrations. Philos. Trans. R. Soc. A 373, 20140404 (2015)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Peschek, E., Boivin, N., Pierre, C., Shaw, S.W.: Nonlinear modal analysis of structural systems using multi-mode invariant manifolds. Nonlinear Dyn. 25, 183–205 (2001)Google Scholar
  32. 32.
    Pesheck, E., Pierre, C., Shaw, S.W.: A new Galerkin-based approach for accurate non-linear normal modes through invariant manifolds. J. Sound Vib. 249(5), 971–993 (2002)Google Scholar
  33. 33.
    Peeters, M., Viguié, R., Sérandour, G., Kerschen, G., Golinval, J.C.: Nonlinear normal modes. Part II: toward a practical computation using numerical continuation techniques. Mech. Syst. Signal Process. 23, 195–216 (2009)Google Scholar
  34. 34.
    Poincaré, J.H.: Sur les propriétés des fonctions définies par les équations différences. Gauthier-Villars, Paris (1879)Google Scholar
  35. 35.
    Redkar, S., Sinha, S.C.: A direct approach to order reduction of nonlinear systems subjected to external periodic excitations. J. Comput. Nonlinear Dyn. 3, 031011-1 (2008)Google Scholar
  36. 36.
    Renson, L., Delíege, G., Kerschen, G.: An effective finite-element-based method for the computation of nonlinear normal modes of nonconservative systems. Meccanica 49, 1901–1916 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Renson, L., Kerschen, G., Cochelin, G.: Numerical computation of nonlinear normal modes in mechanical engineering. J. Sound Vib. 364, 177–206 (2016)CrossRefGoogle Scholar
  38. 38.
    Rosenberg, R.M.: The normal modes of nonlinear \(n\)-degree-of-freedom systems. J. Appl. Mech. 30, 7–14 (1962)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Shaw, S.W., Pierre, C.: Normal modes for non-linear vibratory systems. J. Sound Vib. 164, 85–124 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Shaw, S.W., Pierre, C.: Normal modes of vibration for nonlinear continuous systems. J. Sound Vib. 169, 319–347 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Shaw, S.W., Peschek, E., Pierre, C.: Modal analysis-based reduced-order models for nonlinear structures: an invariant manifold approach. Shock Vib. Dig. 31, 1–16 (1999)CrossRefGoogle Scholar
  42. 42.
    Sinha, S.C., Redkar, S., Butcher, E.A.: Order reduction of nonlinear systems with time periodic coefficients using invariant manifolds. J. Sound Vib. 284, 985–1002 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Sternberg, S.: Local contractions and a theorem of Poincaré. Am. J. Math. 79, 809–824 (1957)MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Vakakis, A. (ed.): Normal Modes and Localization in Nonlinear Systems. Kluwer, Dordrecht (2001)Google Scholar
  45. 45.
    Verhulst, F.: Profits and pitfalls of timescales in asymptotics. SIAM Rev. 57(2), 255–274 (2015)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  1. 1.Institute for Mechanical SystemsETH ZürichZurichSwitzerland

Personalised recommendations