Nonlinear Dynamics

, Volume 70, Issue 3, pp 1931–1949

An upper bound for validity limits of asymptotic analytical approaches based on normal form theory

  • Claude-Henri Lamarque
  • Cyril Touzé
  • Olivier Thomas
Original Paper

DOI: 10.1007/s11071-012-0584-y

Cite this article as:
Lamarque, CH., Touzé, C. & Thomas, O. Nonlinear Dyn (2012) 70: 1931. doi:10.1007/s11071-012-0584-y


Perturbation methods are routinely used in all fields of applied mathematics where analytical solutions for nonlinear dynamical systems are searched. Among them, normal form theory provides a reliable method for systematically simplifying dynamical systems via nonlinear change of coordinates, and is also used in a mechanical context to define Nonlinear Normal Modes (NNMs). The main recognized drawback of perturbation methods is the absence of a criterion establishing their range of validity in terms of amplitude. In this paper, we propose a method to obtain upper bounds for amplitudes of changes of variables in normal form transformations. The criterion is tested on simple mechanical systems with one and two degrees-of-freedom, and for complex as well as real normal form. Its behavior with increasing order in the normal transform is established, and comparisons are drawn between exact solutions and normal form computations for increasing levels of amplitudes. The results clearly establish that the criterion gives an upper bound for validity limit of normal transforms.


Normal form theoryUpper bound of validity limitPerturbation methodsNonlinear Normal Modes

Copyright information

© Springer Science+Business Media B.V. 2012

Authors and Affiliations

  • Claude-Henri Lamarque
    • 1
  • Cyril Touzé
    • 2
  • Olivier Thomas
    • 3
  1. 1.Université de LyonENTPE/DGCB/FRE CNRS 3237Vaulx-en-Velin CedexFrance
  2. 2.Unité de Mécanique (UME)ENSTA—ParisTechParisFrance
  3. 3.Structural Mechanics and Coupled Systems LaboratoryCNAMParisFrance