2023/01/19 by Gergely Buza, Buza, Gergely · 2 citations
Computer Science · Physics and Astronomy · #37L25 #37L65 #Analysis of PDEs (math.AP) #Dynamical Systems (math.DS) #FOS: Mathematics #FOS: Physical sciences #Fluid Dynamics (physics.flu-dyn) #Model Reduction and Neural Networks #Nonlinear Dynamics and Pattern Formation #Quantum chaos and dynamical systems
paper · pdf · doi:10.48550/arxiv.2301.07898
openalex publication_date 2023/01/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Spectral subspaces of a linear dynamical system identify a large class of invariant structures that highlight/isolate the dynamics associated to select subsets of the spectrum. The corresponding notion for nonlinear systems is that of spectral submanifolds -- manifolds invariant under the full nonlinear dynamics that are determined by their tangency to spectral subspaces of the linearized system. In light of the recently-emerged interest in their use as tools in model reduction, we propose an extension of the relevant theory to the realm of fluid dynamics. We show the existence of a large (and the most pertinent) subclass of spectral submanifolds and foliations - describing the behaviour of nearby trajectories - about fixed points and periodic orbits of the Navier-Stokes equations. Their uniqueness and smoothness properties are discussed in detail, due to their significance from the perspective of model reduction. The machinery is then put to work via a numerical algorithm developed along the lines of the parameterization method, that computes the desired manifolds as power series expansions. Results are shown within the context of 2D channel flows.