Beltrami fields exhibit knots and chaos almost surely
Beltrami fields exhibit knots and chaos almost surely
复制标题
贝尔特拉米田几乎肯定会出现结节和混乱
DOI：
10.1017/fms.2023.52
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发表时间：
2020
期刊：
影响因子：
--
通讯作者：
'Alvaro Romaniega
中科院分区：
文献类型：
--
作者：
A. Enciso;D. Peralta;'Alvaro Romaniega
摘要本文证明，在概率
一美元
随机Beltrami场的混沌区域与复杂拓扑的不变环面共存。考虑这个问题的动机是在三维定常欧拉流的研究中产生的。Arnold在1965年推测，一个典型的贝尔特拉米场表现出与一个具有两个自由度的一般哈密顿系统的能量超曲面的限制相同的复杂性。证明取决于高斯随机Beltrami场表现出的马蹄形、零点和打结不变环面和周期轨迹的数量的渐近界的获得，我们通过对高斯随机单色波的Nazarov-Sodin理论的非平凡扩展和动力系统理论的不同工具的应用，包括Kolmogorov-Arnold-Moser（KAM）理论，Melnikov分析与双曲性我们的结果在Beltrami场的情况下，
${\mathbb {R}}^3$
和高频贝尔特拉米场的3-torus。
Abstract In this paper, we show that, with probability
$1$
, a random Beltrami field exhibits chaotic regions that coexist with invariant tori of complicated topologies. The motivation to consider this question, which arises in the study of stationary Euler flows in dimension 3, is V.I. Arnold’s 1965 speculation that a typical Beltrami field exhibits the same complexity as the restriction to an energy hypersurface of a generic Hamiltonian system with two degrees of freedom. The proof hinges on the obtention of asymptotic bounds for the number of horseshoes, zeros and knotted invariant tori and periodic trajectories that a Gaussian random Beltrami field exhibits, which we obtain through a nontrivial extension of the Nazarov–Sodin theory for Gaussian random monochromatic waves and the application of different tools from the theory of dynamical systems, including Kolmogorov–Arnold–Moser (KAM) theory, Melnikov analysis and hyperbolicity. Our results hold both in the case of Beltrami fields on
${\mathbb {R}}^3$
and of high-frequency Beltrami fields on the 3-torus.
影响因子：
3
作者：
Canzani, Yaiza;Sarnak, Peter
通讯作者：
Sarnak, Peter