Question:
Do you use POD to identify a coordinate frame which captures most important coherent structures?
Answer:
Yes, sometime we do find Proper Orthogonal Decomposition (construction of orthogonal coordinate frames by fitting long-time trajectory averages by ellipsoidal cigars) helpful in identifying important directions in the state space, and we are indebted to the earlier POD work for the idea that the energy norm is a useful norm.

 
However, triangulation of state space by exact invariant solutions and their stable/unstable manifolds is a very different idea. We do not average over the turbulent flow; we use exact solutions of Navier-Stokes to identify important structures in fluid flow and chart out the state space.