Principal Lyapunov exponents and principal Floquet spaces of positive random dynamical systems. II. Finite-dimensional systems

TL;DR

This paper applies the general theory of principal Lyapunov exponents and Floquet spaces to finite-dimensional positive random dynamical systems, demonstrating the existence of measurable principal structures under broad conditions.

Contribution

It extends the theory to finite-dimensional systems, showing the existence of principal Floquet subspaces and Lyapunov exponents for positive matrix and cooperative ODE systems.

Findings

Existence of measurable principal Floquet subspaces.

Existence of generalized principal Lyapunov exponents.

Establishment of generalized exponential separations.

Abstract

This is the second part in a series of papers concerned with principal Lyapunov exponents and principal Floquet subspaces of positive random dynamical systems in ordered Banach spaces. The current part focuses on applications of general theory, developed in the authors' paper "Principal Lyapunov exponents and principal Floquet spaces of positive random dynamical systems. I. General theory," Trans. Amer. Math. Soc., in press, to positive random dynamical systems on finite-dimensional ordered Banach spaces. It is shown under some quite general assumptions that measurable linear skew-product semidynamical systems generated by measurable families of positive matrices and by strongly cooperative or type-K strongly monotone systems of linear ordinary differential equations admit measurable families of generalized principal Floquet subspaces, generalized principal Lyapunov exponents, and generalized exponential separations.