Related papers: How to Compute Invariant Manifolds and their Reduc…
Invariant manifolds provide the geometric structures for describing and understanding dynamics of nonlinear systems. The theory of invariant manifolds for both finite and infinite dimensional autonomous deterministic systems, and for…
We introduce a nonlinear stochastic model reduction technique for high-dimensional stochastic dynamical systems that have a low-dimensional invariant effective manifold with slow dynamics, and high-dimensional, large fast modes. Given only…
We present the Constructive Methods of Invariant Manifolds for model reduction in physical and chemical kinetics, developed during last two decades. The problem of reduced description is studied as a problem of constructing the slow…
This paper aims at reviewing nonlinear methods for model order reduction of structures with geometric nonlinearity, with a special emphasis on the techniques based on invariant manifold theory. Nonlinear methods differ from linear based…
The invariant manifold approach is used to explore the dynamics of a nonlinear rotor, by determining the nonlinear normal modes, constructing a reduced order model and evaluating its performance in the case of response to an initial…
Model reduction of high-dimensional dynamical systems alleviates computational burdens faced in various tasks from design optimization to model predictive control. One popular model reduction approach is based on projecting the governing…
We show how spectral submanifold theory can be used to construct reduced-order models for harmonically excited mechanical systems with internal resonances. Efficient calculations of periodic and quasi-periodic responses with the…
Spectral submanifolds (SSMs) have recently been shown to provide exact and unique reduced-order models for nonlinear unforced mechanical vibrations. Here we extend these results to periodically or quasiperiodically forced mechanical…
Random invariant manifolds are geometric objects useful for understanding complex dynamics under stochastic influences. Under a nonuniform hyperbolicity or a nonuniform exponential dichotomy condition, the existence of random pseudo-stable…
This paper investigates model-order reduction methods for geometrically nonlinear structures. The parametrisation method of invariant manifolds is used and adapted to the case of mechanical systems expressed in the physical basis, so that…
A key issue in dimension reduction of dissipative dynamical systems with spectral gaps is the identification of slow invariant manifolds. We present theoretical and numerical results for a variational approach to the problem of computing…
Recently developed reduced-order modeling techniques aim to approximate nonlinear dynamical systems on low-dimensional manifolds learned from data. This is an effective approach for modeling dynamics in a post-transient regime where the…
Invariant manifolds facilitate the understanding of nonlinear stochastic dynamics. When an invariant manifold is represented approximately by a graph for example, the whole stochastic dynamical system may be reduced or restricted to this…
This paper presents a novel non-linear model reduction method: Probabilistic Manifold Decomposition (PMD), which provides a powerful framework for constructing non-intrusive reduced-order models (ROMs) by embedding a high-dimensional system…
Very high dimensional nonlinear systems arise in many engineering problems due to semi-discretization of the governing partial differential equations, e.g. through finite element methods. The complexity of these systems present…
In a nonlinear oscillatory system, spectral submanifolds (SSMs) are the smoothest invariant manifolds tangent to linear modal subspaces of an equilibrium. Amplitude-frequency plots of the dynamics on SSMs provide the classic backbone curves…
Invariant manifolds are fundamental tools for describing and understanding nonlinear dynamics. In this paper, we present a theory of stable and unstable manifolds for infinite dimensional random dynamical systems generated by a class of…
For general nonlinear mechanical systems, we derive closed-form, reduced-order models up to cubic order based on rigorous invariant manifold results. For conservative systems, the reduction is based on Lyapunov Subcenter Manifold (LSM)…
Input-affine dynamical systems often arise in control and modeling scenarios, such as the data-driven case when state-derivative observations are recorded under bounded noise. Common tasks in system analysis and control include optimal…
We show how spectral submanifold (SSM) theory can be used to extract forced-response curves, including isolas, without any numerical simulation in high-degree-of-freedom, periodically forced mechanical systems. We use multivariate…