Majda A J, Timofeyev I
Courant Institute of Mathematical Sciences, New York University, New York, NY 10012, USA.
Proc Natl Acad Sci U S A. 2000 Nov 7;97(23):12413-7. doi: 10.1073/pnas.230433997.
A simplified one-dimensional model system is introduced and studied here that exhibits intrinsic chaos with many degrees of freedom as well as increased predictability and slower decay of correlations for the large-scale features of the system. These are important features in common with vastly more complex problems involving climate modeling or molecular biological systems. This model is a suitable approximation of the Burgers-Hopf equation involving Galerkin projection on Fourier modes. The model has a detailed mathematical structure that leads to a well-defined equilibrium statistical theory as well as a simple scaling theory for correlations. The numerical evidence presented here strongly supports the behavior predicted from these statistical theories. Unlike the celebrated dissipative and dispersive approximations of the Burgers-Hopf equation, which exhibit exactly solvable and/or completely integrable behavior, these model approximations have strong intrinsic chaos with ergodic behavior.
本文介绍并研究了一个简化的一维模型系统，该系统表现出具有多个自由度的内在混沌，以及系统大规模特征的可预测性增加和相关性衰减减慢。这些是与涉及气候建模或分子生物系统的极其复杂得多的问题共有的重要特征。该模型是涉及傅里叶模式上伽辽金投影的伯格斯 - 霍普夫方程的合适近似。该模型具有详细的数学结构，导致定义明确的平衡统计理论以及相关性的简单标度理论。这里给出的数值证据有力地支持了这些统计理论所预测的行为。与著名的伯格斯 - 霍普夫方程的耗散和色散近似不同，后者表现出精确可解和/或完全可积行为，这些模型近似具有具有遍历行为的强内在混沌。
