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

TL;DR

This paper explores two approaches to establish exponential separation of type II in random delay differential equations, using generalized Floquet subspaces and Oseledets decomposition under various conditions.

Contribution

It introduces two distinct methods to prove exponential separation of type II in random delay systems, expanding the theoretical framework and including cases with non-normal cones.

Findings

Existence of generalized Floquet subspaces under cooperativity and irreducibility.

Exponential separation of type II achieved with separable dual space.

Application of Oseledets decomposition when dual space is not separable.

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.