线性乘性噪声破坏三维 Kolmogorov 系统的二维吸引紧致流形
Linear multiplicative noise destroys a two-dimensional attractive compact manifold of three-dimensional Kolmogorov systems
AI总结：
本文研究了三维 Kolmogorov 系统的二维不变球面，建立了其全局吸引准则，并证明线性乘性 Wiener 噪声能破坏该球面并引发平稳测度的分岔。
摘要：
本文首先刻画在 $\mathbb{R}^3$ 中具有二维不变球面的三维 Kolmogorov 系统，随后建立该不变球面在 $\mathbb{R}^3$ 中除原点外的全局吸引准则，并给出球面上带有孤立平衡点的全局动力学。最后，考虑在线性乘性 Wiener 噪声诱导的扰动下吸引不变球面的持久性。结果表明，适当的噪声强度能破坏该球面并导致平稳测度的分岔。
英文摘要：
In the paper we first characterize three-dimensional Kolmogorov systems possessing a two-dimensional invariant sphere in $\mathbb{R}^3$, then establish a global attracting criterion for this invariant sphere in $\mathbb{R}^3$ except the origin, and give global dynamics with isolated equilibria on the sphere. Finally, we consider the persistence of the attractive invariant sphere under the perturbation induced by linear multiplicative Wiener noise. It is shown that suitable noise intensity can destroy the sphere and lead to bifurcation of stationary measures.
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