在本文中，我们进一步开发了一种方法，用于检测不稳定的周期轨道（UPOs）的稳定化变换，其中的策略是将系统的利益，在这样一种方式，轨道变得稳定。使用这种方法的主要困难在于，当我们向更高的维度移动时，过去使用的变换的数量变得势不可挡（Davidchack and Lai 1999; Schmelcher et al. 1997，1998）。我们最近提出了一组稳定化变换，它是从一小部分已经发现的UPO构造的（Crofts和Davidchack 2006）。使用建议的集合的主要好处是，它的基数取决于不稳定流形在UPO的尺寸，而不是系统的尺寸。在一个典型的情况下，不稳定流形的维数比系统的维数小得多，因此变换的数量也小得多。在这里，我们将这种方法扩展到高维系统的常微分方程，并将其应用到一个混沌空间扩展系统的模型例子-Kuramoto-Sivashinsky方程。与Newton-Armijo（NA）和Levernberg-Marquardt（LM）的方法进行了比较。在后一种情况下，我们利用LM算法能够求解欠定方程组的事实，从而消除了对任何额外约束的需要。
英文摘要：
In this paper we develop further a method for detecting unstable periodic orbits (UPOs) by stabilising transformations, where the strategy is to transform the system of interest in such a way that the orbits become stable. The main difficulty of using this method is that the number of transformations, which were used in the past, becomes overwhelming as we move to higher dimensions (Davidchack and Lai 1999; Schmelcher et al. 1997, 1998). We have recently proposed a set of stabilising transformations which is constructed from a small set of already found UPOs (Crofts and Davidchack 2006). The main benefit of using the proposed set is that its cardinality depends on the dimension of the unstable manifold at the UPO rather than the dimension of the system. In a typical situation the dimension of the unstable manifold is much smaller than the dimension of the system so the number of transformations is much smaller. Here we extend this approach to high-dimensional systems of ODEs and apply it to the model example of a chaotic spatially extended system -- the Kuramoto-Sivashinsky equation. A comparison is made between the performance of this new method against the competing methods of Newton-Armijo (NA) and Levernberg-Marquardt (LM). In the latter case, we take advantage of the fact that the LM algorithm is able to solve under-determined systems of equations, thus eliminating the need for any additional constraints.
Inviscid scaling in the Kuramoto-Sivashinsky equation from functional renormalization group and direct numerical simulations
Univ. Grenoble Alpes（格勒诺布尔阿尔卑斯大学）；CNRS（法国国家科学研究中心）；LPMMC（拉萨路斯-庞加莱凝聚态物质与复杂系统实验室）；School of Physics, Indian Institute of Science Education and Research Thiruvananthapuram（印度提鲁瓦南塔普拉姆科学教育与研究所物理学院）；Instituto de Física, Facultad de Ingeniería, Universidad de la República（乌拉圭共和国大学工程学院物理研究所）
