K -dimensional invariant cones of random dynamical systems in RnRn with applications

We investigated a linear random dynamical system which strongly preserves a cone C of dimension-k (abbr. k  -cone) in RnRn. Under some general assumptions, it is shown that such system admits a measurable family of k  -dimensional subspaces and a measurable family of (n−k)(n−k)-dimensional subspaces which are complementary to each other and form into a tempered invariant splitting of RnRn. We further apply the measurable bundles so obtained to study the linear random monotone cyclic feedback systems, as well as the linear competitive–cooperative tridiagonal systems. This generalizes the Floquet theory for these deterministic non-autonomous (or time-periodic) systems to the random systems.