Principal Floquet subspaces and exponential separations of type Ⅱ with applications to random delay differential equations

Janusz Mierczyński,Rafael Obaya,Sylvia Novo

doi:10.3934/dcds.2018265

Abstract

This paper deals with the study of principal Lyapunov exponents, principal Floquet subspaces, and exponential separation for positive random linear dynamical systems in ordered Banach spaces. The main contribution lies in the introduction of a new type of exponential separation, called of type II, important for its application to nonautonomous random differential equations with delay. Under weakened assumptions, the existence of an exponential separation of type II in an abstract general setting is shown, and an illustration of its application to dynamical systems generated by scalar linear random delay differential equations with finite delay is given.

Highlights

This paper continues the study of the existence of principal Lyapunov exponents, principal Floquet subspaces and generalized exponential separations for positive random linear skew-product semiflows in ordered Banach spaces
The concept of generalized exponential separation of type II is introduced as a natural modification of the classical concept, to later show the applicability of this new theory in the context of nonautonomous functional differential equations with finite delay
Mierczynski and Shen [19] provided the assumptions required for general random positive linear skew-product semiflows in order to admit generalized principal Floquet subspaces and generalized exponential separation of type I

Summary

Introduction

This paper continues the study of the existence of principal Lyapunov exponents, principal Floquet subspaces and generalized exponential separations for positive random linear skew-product semiflows in ordered Banach spaces. Mierczynski and Shen [19] provided the assumptions required for general random positive linear skew-product semiflows (with both discrete and continuous time) in order to admit generalized principal Floquet subspaces and generalized exponential separation of type I (in contrast to [17, 18], no embedding into topological semiflows was used in the proofs) The application of this theory to a variety of random dynamical systems arising from Leslie matrix models, cooperative linear ordinary differential equations and linear parabolic partial differential equations can be found in Mierczynski and Shen [20, 21].

Preliminaries

Generalized principal Floquet subspaces

Generalized exponential separation

Findings

Scalar linear random delay differential equations