Invariant manifolds are important constructs for the quantitative and
qualitative understanding of nonlinear phenomena in dynamical systems. In
nonlinear damped mechanical systems, for instance, spectral submanifolds have
emerged as useful tools for the computation of forced response curves, backbone
curves, detached resonance curves (isolas) via exact reduced-order models. For
conservative nonlinear mechanical systems, Lyapunov subcenter manifolds and
their reduced dynamics provide a way to identify nonlinear amplitude-frequency
relationships in the form of conservative backbone curves. Despite these
powerful predictions offered by invariant manifolds, their use has largely been
limited to low-dimensional academic examples. This is because several
challenges render their computation unfeasible for realistic engineering
structures described by finite-element models. In this work, we address these
computational challenges and develop methods for computing invariant manifolds
and their reduced dynamics in very high-dimensional nonlinear systems arising
from spatial discretization of the governing partial differential equations. We
illustrate our computational algorithms on finite-element models of mechanical
structures that range from a simple beam containing tens of degrees of freedom
to an aircraft wing containing more than a hundred-thousand degrees of freedom.

本文研究了在非线性动力学系统中计算不变流形的方法，利用高维有限元模型和 Lyapunov 子中心流形来解决这个问题。

如何计算高维有限元模型中的不变流形及其约化动力学

How to Compute Invariant Manifolds and their Reduced Dynamics in  High-Dimensional Finite-Element Models

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.

本文探讨了非线性正常模 (NNMs) 和 Koopman 算子的特定本征函数的零级集之间存在的关系，建立了一种新的不受折叠影响的不变流形全局参数化方法。通过一个双自由度立体非线性结构的系统，说明了提出的方法。