Discontinuity III

 

Crossroads

I went down to the crossroads
Fell down on my knees
Down to the crossroads
Fell down on my knees
Asked the Lord above for mercy
Take me, if you please

A discontinuity is a crossroads

In a speed-volume curve of traffic or water flow, there is a discontinuity at which laminar/uncongested/orderly flow becomes turbulent/congested/chaotic flow. At this discontinuity there is a crossroads in the behavior of the flow before the discontinuity and the behavior of the flow after the discontinuity. The flow approaches the crossroads/discontinuity from reality, from Point D in the figure below.

At the discontinuity, Point E, it can remain in reality on the left side of the figure, and move to the curve connecting Point A with Point E, from the orange curve to the blue curve. However that requires a rotation by 3/2 π  and this requires that the equation describing the flow from Point D to E to become the logarithm of a negative number, and that behavior is undefined. It can move to the curve connecting Point E to Point at C, also moving from the orange curve to the blue curve, a mirror of the original behavior, but that requires moving the wrong way on a one-way link. It can move to the curve connecting point E with Point B and stay on the orange curve. That is a continuation of the original path, but it requires moving to the right side of the figure, which is NOT observable reality. It is unobservable behavior. At the crossroads any of those behaviors is possible, but each has problems. But moving from reality to unobservable flow seems more likely at the crossroads than any of the other possibilities.