Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Spectral comparison of compound cocycles generated by delay equations in Hilbert spaces

Mikhail Anikushin

TL;DR

The paper develops a spectral comparison framework for compound cocycles generated by nonautonomous delay equations in Hilbert spaces, extending to exterior powers and linking to stationary semigroups via a frequency theorem. It introduces a structural Cauchy formula and embracing-space machinery to handle unbounded perturbations, enabling robust criteria that preserve uniform exponential dichotomies and potentially guarantee global stability. Central to the approach are additive compounds $A^{[\otimes m]}$ and $A^{[\wedge m]}$, their description in terms of diagonal Sobolev/embracing spaces, and the formulation of frequency inequalities using the transfer operator $W(p)$. The results provide a rigorous pathway to exclude closed invariant contours on attractors and suggest practical algorithms for nonlinear delay systems, with computational strategies and scalar-delay exemplars discussed as future work.

Abstract

We study linear cocycles generated by nonautonomous delay equations in a suitable Hilbert space and their extensions, called compound cocycles, to exterior powers. Using a recent version of the frequency theorem, we develop analytical techniques for comparing spectral properties, such as uniform exponential dichotomies, between such cocycles and semigroups generated by stationary equations. These methods are based on properties related to regularity and structure in PDEs associated with delay equations. In particular, the developed machinery leads to effective robust criteria that guarantee the absence of closed invariant contours on global attractors arising in nonlinear problems and are expected to ensure global stability.

Spectral comparison of compound cocycles generated by delay equations in Hilbert spaces

TL;DR

The paper develops a spectral comparison framework for compound cocycles generated by nonautonomous delay equations in Hilbert spaces, extending to exterior powers and linking to stationary semigroups via a frequency theorem. It introduces a structural Cauchy formula and embracing-space machinery to handle unbounded perturbations, enabling robust criteria that preserve uniform exponential dichotomies and potentially guarantee global stability. Central to the approach are additive compounds and , their description in terms of diagonal Sobolev/embracing spaces, and the formulation of frequency inequalities using the transfer operator . The results provide a rigorous pathway to exclude closed invariant contours on attractors and suggest practical algorithms for nonlinear delay systems, with computational strategies and scalar-delay exemplars discussed as future work.

Abstract

We study linear cocycles generated by nonautonomous delay equations in a suitable Hilbert space and their extensions, called compound cocycles, to exterior powers. Using a recent version of the frequency theorem, we develop analytical techniques for comparing spectral properties, such as uniform exponential dichotomies, between such cocycles and semigroups generated by stationary equations. These methods are based on properties related to regularity and structure in PDEs associated with delay equations. In particular, the developed machinery leads to effective robust criteria that guarantee the absence of closed invariant contours on global attractors arising in nonlinear problems and are expected to ensure global stability.
Paper Structure (20 sections, 57 theorems, 284 equations, 1 figure)

This paper contains 20 sections, 57 theorems, 284 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Historical perspective: Lyapunov dimension and effective dimension estimates for delay equations
  3. Contribution of the present work
  4. Structure of the present work
  5. Multiplicative compounds on tensor products of Hilbert spaces
  6. Cocycles, $C_{0}$-semigroups, and additive compounds
  7. Description of additive compounds for delay equations
  8. Structural Cauchy formula for linear inhomogeneous problems
  9. Nonautonomous perturbations of additive compounds for delay equations
  10. Infinitesimal description of compound cocycles
  11. Associated linear inhomogeneous problems with quadratic constraints
  12. Properties of the complexificated problem
  13. Frequency inequalities for spectral comparison
  14. Discussion
  15. Diagonal translation semigroups
  16. ...and 5 more sections

Key Result

Theorem 2.1

For the above defined $L_{2}$-spaces $\mathbb{H}_{1}$ and $\mathbb{H}_{2}$, the mapping where $(\phi_{1} \otimes \phi_{2})(x_{1},x_{2})\coloneq \phi_{1}(x_{1}) \otimes \phi_{2}(x_{2})$ for $(\mu_{1} \otimes \mu_{2})$-almost all $(x_{1},x_{2}) \in \mathcal{X}_{1} \times \mathcal{X}_{2}$, induces an isometric isomorphism between $\mathbb{H}_{1} \otimes \mathbb{H}_{2}$ and $L_{2}(\mathcal

Figures (1)

  • Figure 1: An illustration to the decomposition of $L_{2}([-\tau,0]^{2};\mu^{\otimes 2};\mathbb{R})$ according to \ref{['EQ: TensorSpaceDelayCompoundDecompositionBoundarySubspaces']}, where the restriction operators $R^{(2)}_{0}$, $R^{(2)}_{1}$, $R^{(2)}_{2}$, and $R^{(2)}_{12}$ provide natural isometric isomorphisms between the boundary subspaces over the faces $\mathcal{B}^{(2)}_{0}$, $\mathcal{B}^{(2)}_{1}$, $\mathcal{B}^{(2)}_{2}$, and $\mathcal{B}^{(2)}_{12}$ and appropriate $L_{2}$-spaces respectively.

Theorems & Definitions (117)

  • Theorem 2.1
  • proof
  • Theorem 2.2: BrownPearcy1966
  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Theorem 2.3
  • proof
  • Proposition 2.1
  • Proposition 3.1
  • ...and 107 more