Question:
Why “lower branch”, “upper branch?”
Answer:
The drag I exerted on the walls and the bulk dissipation rate D are physical quantities associated with a plane Couette fluid state.

 
[Reynolds/dissipation plot]
Bifurcation diagram
blue dots
Upper branch equilibrium EQ2. More structure, hence higher dissipation.
blue circles
Lower branch equilibrium EQ1. Smooth, hence low dissipation.
Saddle-node bifurcation at Re = 218.5... gives birth to the Nagata lower and upper branches. Also indicated: Dissipation as a function of Reynolds number for other equilibria and traveling waves. All solutions in this tutorial are studied at Re=400.