2026/07/24 by Bochun Chang, Xiaofang Lin, Yi Wang
#math.DS #math.PR
We focus on a stochastic system that models a collection of N identical oscillators with unidirectional coupling, perturbed by white noise. The unidirectional coupling breaks the symmetry of mutual dependence among the oscillators. It is shown that the associated random dynamical system ϕ admits a dynamical order, meaning that ϕ admits a simple asymptotic one-dimensional structure that is totally ordered with respect to the standard order in ℝN. Our approach takes a novel geometric perspective: we introduce a random dynamical system Φ on a smooth Riemannian manifold M, diffeomorphic to \mathbbS1 × ℝN-1, by ``wrapping'' the random system ϕ from ℝN onto M. By choosing an appropriate random cone field CM on M, we show that Φ is a differentially positive random system on M, which is a random counterpart of the differentially positive systems introduced by Forni and Sepulchre. We further demonstrate that the traditional well-known horizontal curves can be identified as the conal curves on M, thereby providing a crucial tool for establishing the dynamical order of the random system ϕ.