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Existence of spectral submanifolds in time delay systems

Gergely Buza, George Haller

TL;DR

This work develops a rigorous spectral-submanifold (SSM) theory for time-delay equations within the sun-star framework, establishing local invariant manifolds tangent to spectral subspaces and global inertial manifolds under a sufficient spectral gap. It shows how delay-induced spectral structure yields both local SSMs and, under global Lipschitz and large-gap conditions, globally attracting inertial manifolds whose dimension need not equal the physical space dimension. The authors provide explicit reduction mechanisms, including a graph-parametrization approach and higher-order expansions for the reduced dynamics, and demonstrate the approach via concrete delay-differential equation examples (e.g., a Cushing-type equation and a 1D scalar delay). A key practical insight is that shrinking the delay can enlarge the spectral gap, enabling smoother, faster attraction to IMs and recovering classical small-delay results in a unified framework. The results pave the way for equation- and data-driven reduced models of complex delay systems with provable convergence, smoothness, and attractivity properties, including foliations that describe convergence toward the reduced dynamics.

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.

Existence of spectral submanifolds in time delay systems

TL;DR

This work develops a rigorous spectral-submanifold (SSM) theory for time-delay equations within the sun-star framework, establishing local invariant manifolds tangent to spectral subspaces and global inertial manifolds under a sufficient spectral gap. It shows how delay-induced spectral structure yields both local SSMs and, under global Lipschitz and large-gap conditions, globally attracting inertial manifolds whose dimension need not equal the physical space dimension. The authors provide explicit reduction mechanisms, including a graph-parametrization approach and higher-order expansions for the reduced dynamics, and demonstrate the approach via concrete delay-differential equation examples (e.g., a Cushing-type equation and a 1D scalar delay). A key practical insight is that shrinking the delay can enlarge the spectral gap, enabling smoother, faster attraction to IMs and recovering classical small-delay results in a unified framework. The results pave the way for equation- and data-driven reduced models of complex delay systems with provable convergence, smoothness, and attractivity properties, including foliations that describe convergence toward the reduced dynamics.

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.
Paper Structure (33 sections, 34 theorems, 240 equations, 4 figures)

This paper contains 33 sections, 34 theorems, 240 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Overview
  3. Delay equations
  4. DDE as an evolution equation
  5. Properties of semiflows generated by delay equations
  6. Linearization about a stationary state
  7. Spectrum of A
  8. Characteristic equation
  9. Asymptotics of the spectrum, barriers to Ck smoothness of pseudo-unstable manifolds
  10. Statement of results
  11. Inertial manifolds
  12. Examples
  13. The Cushing equation
  14. Small delay inertial manifolds for a one-dimensional example
  15. The example of [SOS25]
  16. ...and 18 more sections

Key Result

Theorem 1

Suppose that the spectral subset $\Sigma$ is of the form $\Sigma = \{\lambda \in \sigma(A) \; | \; \mathrm{Re } \, \lambda \geq \gamma\}$ for some $\gamma \in \mathbb{R}$. Then, there exists a neighbourhood $U$ of the stationary state $0 \in X$ and a locally invariant (under the semiflow $\varphi$) Moreover, $W^\Sigma$ attracts trajectories that remain in $U$ at an exponential rate of $\gamma$, s

Figures (4)

  • Figure 1: Roots of \ref{['eq:charact_cushing']} in the complex plane.
  • Figure 2: The real roots of \ref{['eq:charact_example2']}, obtained as the intersection of $x \mapsto e^{-xh}$ and $x \mapsto -x$ for three different values of $h$. The black dashed line shows $x \mapsto -x$.
  • Figure 3: The graph of \ref{['eq:tau_R']} for two values of $h$.
  • Figure 4: (Left) dependence of of the maximal delay $h_{max}$ on $\beta_2$ -- according to the (suboptimal) prediction of \ref{['eq:h_cond_finale']}. (Right) The graph of \ref{['eq:tau_F']} compared with \ref{['eq:tau_R']} for $h = 0.002$ and $\beta_2 = -500$.

Theorems & Definitions (78)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Proposition 4: Results of Sections VII.3, VII.4 and VII.6, diekmann2012delay
  • proof
  • Theorem 5: Theorem B.IV.2.1, arendt1986one
  • proof
  • Lemma 6: Spectral projections
  • proof
  • Remark 7
  • ...and 68 more