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Two dynamical approaches to the notion of exponential separation for random systems of delay differential equations

Marek Kryspin, Janusz Mierczynski, Sylvia Novo, Rafael Obaya

TL;DR

The paper addresses exponential separation of type II for positive random delay systems by developing two complementary approaches. First, a direct theory of generalized principal Floquet subspaces and type II separation for measurable linear skew-product semiflows on ordered Banach spaces is developed, with integrability, positivity, and focusing conditions guiding the construction; separability of the dual $X^*$ yields a clear type II decomposition. Second, the authors leverage an Oseledets decomposition to realize type II separation when the direct theory requires $X^*$ separability, unifying the two perspectives. The results are then extended to linear random delay systems in spaces $C$ and $L$, with a transfer mechanism via the injection $J$, and finally to non-separable dual settings across three concrete cases, including cones that may be non-normal. Together, these contributions provide a robust framework for confirming generalized exponential separation in a broad class of random delay systems, with concrete implications for numerical computation and analysis of principal Floquet subspaces in non-autonomous settings.

Abstract

This paper deals with the exponential separation of type II, an important concept for random systems of differential equations with delay, introduced in \JM\ et al.~\cite{MiNoOb1}. Two different approaches to its existence are presented. The state space $X$ will be a separable ordered Banach space with $\dim X\geq 2$, dual space $X^{*}$ and positive cone $X^+$ normal and reproducing. In both cases, appropriate cooperativity and irreducibility conditions are assumed to provide a family of generalized Floquet subspaces. If in addition $X^*$ is also separable, one obtains a exponential separation of type II. When this is not the case, but there is an Oseledets decomposition for the continuous semiflow, the same result holds. Detailed examples are given for all the situations, including also a case where the cone is not normal.

Two dynamical approaches to the notion of exponential separation for random systems of delay differential equations

TL;DR

The paper addresses exponential separation of type II for positive random delay systems by developing two complementary approaches. First, a direct theory of generalized principal Floquet subspaces and type II separation for measurable linear skew-product semiflows on ordered Banach spaces is developed, with integrability, positivity, and focusing conditions guiding the construction; separability of the dual yields a clear type II decomposition. Second, the authors leverage an Oseledets decomposition to realize type II separation when the direct theory requires separability, unifying the two perspectives. The results are then extended to linear random delay systems in spaces and , with a transfer mechanism via the injection , and finally to non-separable dual settings across three concrete cases, including cones that may be non-normal. Together, these contributions provide a robust framework for confirming generalized exponential separation in a broad class of random delay systems, with concrete implications for numerical computation and analysis of principal Floquet subspaces in non-autonomous settings.

Abstract

This paper deals with the exponential separation of type II, an important concept for random systems of differential equations with delay, introduced in \JM\ et al.~\cite{MiNoOb1}. Two different approaches to its existence are presented. The state space will be a separable ordered Banach space with , dual space and positive cone normal and reproducing. In both cases, appropriate cooperativity and irreducibility conditions are assumed to provide a family of generalized Floquet subspaces. If in addition is also separable, one obtains a exponential separation of type II. When this is not the case, but there is an Oseledets decomposition for the continuous semiflow, the same result holds. Detailed examples are given for all the situations, including also a case where the cone is not normal.
Paper Structure (12 sections, 31 theorems, 119 equations)

This paper contains 12 sections, 31 theorems, 119 equations.

Table of Contents

  1. Introduction
  2. A direct theory providing generalized exponential separation
  3. Measurable linear skew-product semidynamical systems
  4. Ordered Banach spaces
  5. Assumptions
  6. Generalized exponential separation
  7. Generalized exponential separation as a consequence of Oseledets decomposition
  8. Semiflows generated by linear random delay differential equations
  9. Semiflows generated by linear random delay systems in a state space with not separable dual
  10. Case 1
  11. Case 2
  12. Case 3

Key Result

Theorem 2.5

Under assumptions A1 and A3, there is an invariant set $\bar{\Omega}_1$ and an $(\mathfrak{F},\mathfrak{B}(X))$-measurable function $w\colon\bar{\Omega}_1\to X^+,\;\omega\mapsto w(\omega)$, $\|w(\omega)\|=1$ for each $\omega\in \bar{\Omega}_1$, such that

Theorems & Definitions (66)

  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Definition 2.4
  • Theorem 2.5
  • proof
  • Definition 2.6
  • Theorem 2.7
  • Theorem 3.1
  • Proposition 3.2
  • ...and 56 more