Heteroclinic connections

Here we are, wandering in 61,506 dimensions. Can one slice up this high-dimensional state space into subregions, develop symbolic dynamics that describes these exact time-invariant solutions of Navier-Stokes equations?
[heteroclinic connection movie of plane Couette flow]

Each heteroclinic connection (see arXiv.org:0808.1865) is traced out by an infinite-time state space trajectory, ie., it corresponds to one 3D flow evolving in time. Dual visualization helps

develop intuition
about how states of different arrangements of rolls and streaks transmogrify into each other
partition state space
into regions tied to each other by robust heteroclinic connections