Attracting Lagrangian Coherent Structures on Riemannian manifolds

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 attracting LCSs with ridges of the backward FTLE. We show that in two-dimensional incompressible flows attracting LCSs appear as ridges of the forward FTLE field, too. This raises the issue of characterization of attracting LCSs from a forward finite-time Lyapunov analysis. To this end, we extend recent results by Haller & Sapsis (2011) [1], regarding the relation 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 give a short and direct proof of the main result of Farazmand & Haller (2013) [2], and prove a new characterization of attracting LCSs in forward time for Haller's variational approach to hyperbolic LCSs [3]. We apply these results to the attracting FTLE ridge of the incompressible saddle flow.