The Birman-Williams theorem gives a connection between the collection of unstable periodic orbits (UPOs) contained within a chaotic attractor and the topology of that attractor, for three-dimensional systems. In certain cases, the fractal dimension of a chaotic attractor in a partial differential equation (PDE) is less than three, even though that attractor is embedded within an infinite-dimensional space. Here we study the Kuramoto-Sivashinsky PDE at the onset of chaos. We use two different dimensionality-reduction techniques -- proper orthogonal decomposition and an autoencoder neural network -- to find two different mappings of the chaotic attractor into three dimensions. By finding the image of the attractor's UPOs in these reduced spaces and examining their linking numbers, we construct templates for the branched manifold which encodes the topological properties of the attractor. The templates obtained using two different dimensionality reduction methods are equivalent. The organization of the periodic orbits is identical and consistent symbolic sequences for low-period UPOs are derived. While this is not a formal mathematical proof, this agreement is strong evidence that the dimensional reduction is robust, in this case, and that an accurate topological characterization of the chaotic attractor of the chaotic PDE has been achieved.
On the use of stabilising transformations for detecting unstable periodic orbits in the Kuramoto-Sivashinsky equation
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。
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）
