Karrasch Daniel
ETH Zürich, Institute for Mechanical Systems, Leonhardstrasse 21, 8092 Zürich, Switzerland.
Chaos. 2015 Aug;25(8):087411. doi: 10.1063/1.4928451.
It is a wide-spread convention to identify repelling Lagrangian Coherent Structures (LCSs) with ridges of the forward finite-time Lyapunov exponent (FTLE) field and to identify attracting LCSs with ridges of the backward FTLE. However, we show that, in two-dimensional incompressible flows, also attracting LCSs appear as ridges of the forward FTLE field. This raises the issue of the characterization of attracting LCSs using a forward finite-time Lyapunov analysis. To this end, we extend recent results regarding the relationship between forward and backward maximal and minimal FTLEs, to both the whole finite-time Lyapunov spectrum and to stretch directions. This is accomplished by considering the singular value decomposition (SVD) of the linearized flow map. By virtue of geometrical insights from the SVD, we provide characterizations of attracting LCSs in forward time for two geometric approaches to hyperbolic LCSs. We apply these results to the attracting FTLE ridge of the incompressible saddle flow.
将排斥性拉格朗日相干结构（LCSs）与正向有限时间李雅普诺夫指数（FTLE）场的脊线相识别，以及将吸引性LCSs与反向FTLE的脊线相识别，这是一种广泛采用的惯例。然而，我们表明，在二维不可压缩流中，吸引性LCSs也表现为正向FTLE场的脊线。这就引发了使用正向有限时间李雅普诺夫分析来表征吸引性LCSs的问题。为此，我们将关于正向和反向最大及最小FTLE之间关系的近期结果扩展到整个有限时间李雅普诺夫谱以及拉伸方向。这是通过考虑线性化流映射的奇异值分解（SVD）来实现的。借助于SVD的几何见解，我们针对双曲LCSs的两种几何方法，给出了正向时间吸引性LCSs的特征描述。我们将这些结果应用于不可压缩鞍点流的吸引性FTLE脊线。
