On the validity limits of the parametrisation method for invariant manifolds: an assessment of practical criteria for vibrating systems
Abstract
The parametrisation method for invariant manifolds is a powerful technique for deriving reducedorder models in the context of nonlinear vibrating systems with geometric nonlinearities, allowing accurate computations of nonlinear normal modes. Thanks to arbitrary order asymptotic expansions, converged results are within reach and directly applicable to finite element structures. However, since it relies on a local theory and asymptotic expansions, the results are only valid up to a given amplitude, which defines the convergence radius of the approximation. The aim of this contribution is to investigate the validity limits of the approach and review the existing error estimates, with the concrete objective of proposing a practical approach to estimate the validity range during the computation, thus producing safe bounds within which the reduced-order model can be used. Three different criteria are assessed. The first one uses the error in the invariance equation as the distance to the fixed point increases. The second one is adapted from an upper bound criterion derived for normal form transforms and based on the potential singularities of the homological operator. The third one uses validity limits of series expansion through Cauchy and d'Alembert rules, which can be tested either on the reduced dynamics coefficients or those of the nonlinear mappings. The criteria are tested on a number of different examples that are representative of the situations encountered when dealing with nonlinear vibrations. The Duffing equation serves as a first benchmark that allows considering conservative oscillations, forced systems at primary resonance, and superharmonic resonance. The investigations are then extended to a vibrating system with two degrees of freedom. Finally, the different criteria are assessed on a finite element beam structure, and guidelines are formulated to generalise their practical use and produce accurate and easy-to-use error bounds in the context of model order reduction for nonlinear vibrating structures.
Related
- Futaki invariant on lcK manifolds with potential
- T^2-invariant Einstein metrics on closed 4-manifolds
- Invariant Manifolds and Stability for Singular SPDEs
- A non-perturbative topological persistence theorem for normally hyperbolic invariant manifolds
- A Note on Partially Anti-invariant Submanifolds of Kenmotsu Manifolds
BibTeX
@article{stabile2026validity,
title = {{On the validity limits of the parametrisation method for invariant manifolds: an assessment of practical criteria for vibrating systems}},
author = {André Stabile and Aurélien Grolet and Alessandra Vizzaccaro and Cyril Touzé},
journal = {Nonlinear Dynamics},
year = {2026},
month = aug,
doi = {10.1007/s11071-026-12956-0},
url = {https://hal.science/hal-05739498v1},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - JOUR
TI - On the validity limits of the parametrisation method for invariant manifolds: an assessment of practical criteria for vibrating systems
AU - André Stabile
AU - Aurélien Grolet
AU - Alessandra Vizzaccaro
AU - Cyril Touzé
PY - 2026
DA - 2026/08/19
JO - Nonlinear Dynamics
DO - 10.1007/s11071-026-12956-0
UR - https://hal.science/hal-05739498v1
AB - The parametrisation method for invariant manifolds is a powerful technique for deriving reducedorder models in the context of nonlinear vibrating systems with geometric nonlinearities, allowing accurate computations of nonlinear normal modes. Thanks to arbitrary order asymptotic expansions, converged results are within reach and directly applicable to finite element structures. However, since it relies on a local theory and asymptotic expansions, the results are only valid up to a given amplitude, which defines the convergence radius of the approximation. The aim of this contribution is to investigate the validity limits of the approach and review the existing error estimates, with the concrete objective of proposing a practical approach to estimate the validity range during the computation, thus producing safe bounds within which the reduced-order model can be used. Three different criteria are assessed. The first one uses the error in the invariance equation as the distance to the fixed point increases. The second one is adapted from an upper bound criterion derived for normal form transforms and based on the potential singularities of the homological operator. The third one uses validity limits of series expansion through Cauchy and d'Alembert rules, which can be tested either on the reduced dynamics coefficients or those of the nonlinear mappings. The criteria are tested on a number of different examples that are representative of the situations encountered when dealing with nonlinear vibrations. The Duffing equation serves as a first benchmark that allows considering conservative oscillations, forced systems at primary resonance, and superharmonic resonance. The investigations are then extended to a vibrating system with two degrees of freedom. Finally, the different criteria are assessed on a finite element beam structure, and guidelines are formulated to generalise their practical use and produce accurate and easy-to-use error bounds in the context of model order reduction for nonlinear vibrating structures.
SN - 0924-090X
SN - 1573-269X
ER -