2025/12/26 by Gergely Buza, George Haller, Buza, Gergely +1
Engineering · #34K19 #Adaptive Control of Nonlinear Systems #Chaotic Dynamics (nlin.CD) #Control Systems and Identification #Dynamical Systems (math.DS) #FOS: Mathematics #FOS: Physical sciences #Stability and Controllability of Differential Equations
paper · doi:10.48550/arxiv.2512.22062
openalex publication_date 2025/12/26 · openalex created_date 2025/12/30 · openalex updated_date 2026/07/28
Spectral submanifolds (SSMs) are invariant manifolds of a dynamical system, defined by the property of being tangent to a spectral subspace of the linearized dynamics at a steady state. We show existence, along with certain desirable properties such as smoothness, attractivity and conditional uniqueness, of SSMs associated to a large class of spectral subspaces in time delay systems. Building on these results, we generalize the criteria for existence of inertial manifolds -- defined as globally exponentially attracting Lipschitz invariant manifolds of finite dimension -- and show that they need not have dimension equal to that of the physical configuration, in contrast to previous accounts. We then demonstrate the applicability of these results on a few simple examples.