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

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 [J. Mierczyǹski, W. Shen, Principal Lyapunov exponents and principal Floquet spaces of positive random dynamical systems, I, general theory, Trans. Amer. Math. Soc., in press (http://dx.doi.org/10.1090/S0002-9947-2013-05814-X). Preprint available at http://arxiv.org/abs/1209.3475], 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.