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====== April 14 study group ====== Some topics I would like to discuss. Feel free to add your own. ===== Poincare sections and periodic orbits ===== Dustin and I have just begun producing Poincare sections of the unstable manifold around the Nagata upper branch equilibrium. These are supposed to help us determine good initial guesses for periodic orbits. Frankly, I don't get how, and I would like some help understanding this. Chaosbook shows how to do with a few low-dimensional examples, by making a Poincare section, looking at the return map of the unstable manifold on the section, and taking approximate fixed points of this map and its iterates as initial guesses for orbits. But the examples, e.g. Rössler, have helpful properties that are not present in plane Couette, namely insanely strong contraction along a 1-dimensional stable direction. So the question is, how can we find approximate fixed points of the return map when there are ten or twenty directions of weaker contraction? Suggested reading: * [[gtspring2009:gibson:w03#poincare_section|Gibson blog on poincare sections]]. * [[http://www.chaosbook.org/chapters/cycles.pdf]] ===== Factoring out the 4th-order translation symmetry =====