Kolmogorov-Lorenz系统的能量学、骨架动力学和长期预测
Energetics, skeletal dynamics and long-term predictions in Kolmogorov-Lorenz systems
摘要：
从能量和Casimir平衡的观点出发，研究了一类低维Kolmogorov-Lorenz系统混沌模型的特殊返回映射，它得到了一个优美的广义Hamilton描述，并包括著名的Lorenz 63情形.特别是它被认为是详细的一个子类，这些模型，正是那些从洛伦兹63通过一个小扰动的标准参数，其中包括例如强迫洛伦兹情况下，在Ref。[6]的文件。本文分为两个部分。在第一部分中，所提到的态函数的极端被认为是，它定义了一个不变的流形，用于构建一个适当的庞加莱曲面为我们的返回映射。从实验观察到的两个不稳定的固定点周围的简单轨道运动的情况下，这些轨道是由它们的能量或Casimir最大值分类，我们构建了一个概念上简单的骨架动力学有效的子类，很好地再现了Lorenz映射Casimir。这种充满活力的方法揭示了一些政权过渡的物理机制。本文的第二部分是专门调查的一种新类型的最大能量为基础的长期预测，其中知识的一个特定的最大能量壳金额的知识的未来（定性）的行为的系统。它表明，在这方面，当地的可预测性分析是不适合这种行为的一个完整的表征。在（地质）流体动力学的更现实的情况下，这种类型的可预测性分析的可能扩展的角度进行了讨论，在论文的最后。
英文摘要：
We study a particular return map for a class of low dimensional chaotic models called Kolmogorov Lorenz systems, which received an elegant general Hamiltonian description and includes also the famous Lorenz63 case, from the viewpoint of energy and Casimir balance. In particular it is considered in detail a subclass of these models, precisely those obtained from the Lorenz63 by a small perturbation on the standard parameters, which includes for example the forced Lorenz case in Ref.[6]. The paper is divided into two parts. In the first part the extremes of the mentioned state functions are considered, which define an invariant manifold, used to construct an appropriate Poincare surface for our return map. From the experimental observation of the simple orbital motion around the two unstable fixed points, together with the circumstance that these orbits are classified by their energy or Casimir maximum, we construct a conceptually simple skeletal dynamics valid within our sub class, reproducing quite well the Lorenz map for Casimir. This energetic approach sheds some light on the physical mechanism underlying regime transitions. The second part of the paper is devoted to the investigation of a new type of maximum energy based long term predictions, by which the knowledge of a particular maximum energy shell amounts to the knowledge of the future (qualitative) behaviour of the system. It is shown that, in this respect, a local analysis of predictability is not appropriate for a complete characterization of this behaviour. A perspective on the possible extensions of this type of predictability analysis to more realistic cases in (geo)fluid dynamics is discussed at the end of the paper.
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