AI 中文总结
研究单向耦合噪声振荡器随机系统，通过将其从\(\mathbb{R}^N\)“包裹”到流形\(M\)引入\(\Phi\)，选合适随机锥场，证明\(\Phi\)是微分正随机系统，借此建立\(\phi\)的动力学序。
AI 中文摘要
我们关注一个随机系统，它模拟了由白噪声扰动的具有单向耦合的\(N\)个相同振荡器的集合。单向耦合打破了振荡器之间相互依赖的对称性。结果表明相关随机动力系统\(\phi\)具有动力学序，即\(\phi\)具有简单的渐近一维结构，相对于\(\mathbb{R}^N\)中的标准序是全序的。我们的方法采用了新颖的几何视角：通过将随机系统\(\phi\)从\(\mathbb{R}^N\)“包裹”到与\(\mathbb{S}^1\times\mathbb{R}^{N - 1}\)微分同胚的光滑黎曼流形\(M\)上，引入一个随机动力系统\(\Phi\)。通过在\(M\)上选择合适的随机锥场\(\mathcal{C}_M\)，我们表明\(\Phi\)是\(M\)上的微分正随机系统，这是Forni和Sepulchre引入的微分正系统的随机对应物。我们进一步证明传统的著名水平曲线可被识别为\(M\)上的锥曲线，从而为建立随机系统\(\phi\)的动力学序提供了关键工具。
英文摘要
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 $\mathbb{R}^N$. Our approach takes a novel geometric perspective: we introduce a random dynamical system $Φ$ on a smooth Riemannian manifold $M$, diffeomorphic to $\mathbb{S}^1 \times \mathbb{R}^{N-1}$, by ``wrapping'' the random system $ϕ$ from $\mathbb{R}^N$ onto $M$. By choosing an appropriate random cone field $\mathcal{C}_M$ 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 $ϕ$.
