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Project: Poincare sections and return maps

most important: Compute an W03 cell unstable manifold Poincaré section and return map for the upper branch solution. Place Gibson's new periodic orbit P47.18 in the W03 cell on it.

I will take the W03 unstable manifold Poincaré section and return map for the upper branch project. I have a feeling that I could readily jump into this one given the tools recently provided in channelflow-1.3.4. — Dustin Spieker 2009-03-02 14:39

Because of inherent symmetries of the problem that create four different representations of the same solution in state space, we need to figure out a way to only count the crossings in one of the four subspaces, probably the canonical (no symmetry operations performed) subspace. This creates a problem if two subsequent time steps cross the poincare section AND from one subspace to another simultaneously. Need to account for this. — Dustin Spieker 2009-03-10 17:00

Right about the time I was finishing my first go around of making a Poincare Section of the UB unstable manifold, I looked at the homepage and saw that you, John G., had finished one. Yours looked much better than the monstrosity my computer spit out. I read over the considerations you made when constructing yours, and realized that I had made almost none, but rather blindly stumbled through the calculation with more or less arbitrary eigenvectors. That's the bad news. The good news is that I understood the considerations you took, and I am ready to give it another shot. — Dustin Spieker 2009-03-18 09:32

Second Try, something is still up, but at least my new plots are somewhat prettier. More on analysis of the plots in a few minutes.

So, I only used the generic poincare.cpp to obtain these results, so none of the symmetries have been factored out. Something is inconsistent between John G's representation and mine of the (u,e0) vs. (u,e1) plot which will need further investigation, and once I put up my method, John G. will be able to shed some light on my problem. I was unable to connect the points together because of how I constructed my data files and fed them into matlab, yet another manifestation of my lack of knowledge in matlab. More to come… — Dustin Spieker 2009-04-07 14:12

My current focuses are now to get eq2poincare.cpp working in synch with channelflow, changing the mixing angle () between e0 and e1 and see how that changes the poincare section, and maybe looking into working with some of the other solutions.

Are you having problems compiling eq2poincare.cpp? Skype me, baby! No cause to worry that you might not have thought of everything on your first shot; i surely haven't either, and two minds will do better than one. John Gibson 2009-05-08 noon EST

Based on the comparative poincare sections I have produced using the eq2poincare.cpp, I don't think that the dynamics should be confined to any symmetry quadrant, but rather allowed to explore all quadrants. The following plots should correspond exactly to each other, except one set was allowed to explore the whole space and the others were reflected so that they would stay in one quadrant. John G. could you please put up the data file for the 47.18 periodic orbit?— Dustin Spieker 2009-05-13 11:25