How to locate the bifurcation point of the flow

As you can see from the figure, this obvious northwest flow will bifurcate. Now I want to find the location of the bifurcation point, and how do I do it ? Of course the flow data are known.

[Sachin Lodhi]

Hello Charles,

Please try to read this. It has all relevant information required for plotting bifurcation points and diagram.

http://www.math.le.ac.uk/people/rld8/ma1251/lab3.html

https://stackoverflow.com/questions/10693525/generate-bifurcation-diagram-for-2d-system

Hope this helps.

Best Regards, Sachin

Answer 1

[William Rose]

@Charles Taylor Taylor,

You provided a quiver plot. Since you have a quiover plot, you obviously have the arrays X,Y,U,V. You can use that data to generate streamlines. Here is an example, using the wind data set.

load wind
X = x(:,:,10);  % get the values from vertical level 10, for 2-D analysis
Y = y(:,:,10);
U = u(:,:,10);
V = v(:,:,10);
quiver(X,Y,U,V); xlabel('x'); ylabel('y')   % make quiver plot
[startX,startY] = meshgrid(X(1,1),Y(:,1));  % start points along left edge
hold on
verts=stream2(X,Y,U,V,startX,startY);       % compute streamlines
streamline(verts)                           % add streamlines to plot

I recommend that you use the streamline data in verts to develop an algorithm to find bifurcation points. This could involve progressively finer searches for nearby starting points that lead to widely differing ending points. Good luck.

[William Rose]

@Charles Taylor Taylor,

In my initial answer, I recommended anaylzing the streamlines, and I explained how to compute the streamlines. I'm sure that a lot of analysis has been done on this topic. I think you will do well to read some papers, rather than try to make up your own approach. Google "bifurcation in a flow filed" or somehting similar. For example, chapter 2 of this PhD thesis has some interesting and potentially relevant ideas.

First you must come up with a mathematcal description of a bifurcation point (or, better yet, find a definition in the literature). Analysis of the velocity field (U,V arrays) could be just as effective as analysis of streamlines. You could search for a place in the velocity field where adjacent velocity vectors point in very different directions. Velocity directions may be unreliable where the velocity is very low, so you may want to account for that.

Good luck.