| Abstract: |
| Dynamical order refers to the property that the global random attractor is a conal curve. It is established herein for noisy oscillator under unidirectional strong coupling on a line. The existence of one-dimensional global random attractor is obtained by identifying it as a manifold via stable foliation theory. We introduce the concept of random differentially positive systems to analyze its order structure. A key difficulty is that the evolution of a conal curve need not converge in general. To address this, we construct a suitably designed cone field that ensures uniform control over the growth of the evolution. By analyzing the evolution of a special conal curve, we prove that its evolution admits a sequential limit in the $C^{0}$-topology and that the global random attractor coincides with the limit, which is itself a closed conal curve, thereby establishing the dynamical order. |
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