4b. Equilibria and their manifolds

While different exact equilibria are clearly physically distinct, it is hard to intuit how are they related from their 3D portraits. The geometry of the transitionally turbulent flows is revealed by their state space portraiture.

State space portrait of 25 exact equilibria and traveling waves of plane Couette flow projected onto a basis constructed from half-cell shifts τ_x, τ_z, and τ_xz of UB .

[from J. Halcrow, J. F. Gibson and P. Cvitanović]

Legend:

The single solid black dot is the laminar state LM.
The black lines touching it are half cell shifts of the lower half of LB (red dots) unstable manifold.
The green dots in between the two are half- cell shifts of NNB.
The path of the streamwise traveling wave TW2 and its countertraveling partner are traced in red.
The spanwise traveling wave, TW_LB and its counter-traveling partnertrace out purple bowties.
The open black circles further out from there are the shifts of NB .
Past those, blue dots are the shifts of EQ5 .

The green dense region past those is an indication of the natural measure generated by a typical long lived turbulent transient and its translations.

Finally, the outermost blue spirals are the fully symmetric portion of UB 's unstable manifold.