Periodic eigendecomposition, to be formulated in this paper, is a numerical method to compute Floquet spectrum and Floquet vectors along periodic orbits in a dynamical system. It is rooted in numerical algorithms advances in computation of 'covariant vectors' of the linearized flow along an ergodic trajectory in a chaotic system. Recent research on covariant vectors strongly strongly suggests that the physical dimension of inertial manifold of a dissipative PDE can be characterized by a finite number of 'entangled modes', dynamically isolated from the residual set of transient degrees of freedom. We anticipate that Floquet vectors display similar properties as covariant vectors. In this paper we incorporate periodic Schur decomposition to the computation of dynamical Floquet vectors, compare it with other methods, and show that the method can yield the full Floquet spectrum of a periodic orbit at every point along the orbit to high accuracy. Its power, and in particular its ability to resolve eigenvalues whose magnitude differs by hundreds of orders magnitude, is demonstrated by applying the algorithm to computation of the full linear stability spectrum of several periodic solutions in one dimensional Kuramoto-Sivashinsky flow.
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）
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。
