Existence of spectral submanifolds in time delay systems

TL;DR

This paper proves the existence and properties of spectral submanifolds in time delay systems, broadening understanding of invariant manifolds and inertial manifolds beyond previous limitations.

Contribution

It establishes the existence, smoothness, and attractivity of spectral submanifolds in time delay systems and generalizes inertial manifold criteria to possibly different dimensions.

Findings

Spectral submanifolds exist with desirable properties in time delay systems.

Inertial manifolds can have dimensions different from the physical configuration.

Results are demonstrated on simple example systems.

Abstract

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.