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====== April 14 study group ======

Some topics I would like to discuss. Feel free to add your own. //John Gibson 2009-04-10 11:11 EST//

===== 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?

8-) I'll try to upload later today the [[http://www.chaosbook.org/chapters/smale.pdf|ChaosBook.org/chapters/smale.pdf]] handcrafted to counter the undue pessimism. Brief version is that the expansion is what matters, contraction is secondary. For this reason my examples in this chapter are billiards and the Hamiltonian Hénon map, the very opposite of insane contraction: they are symplectic, area preserving. 

I do not understand the discontinuities in quotienting C_2, but that is minor - the real problem might be mnore interesting, ie that there is no return map, but only sequences of forward maps (composition of which would then give return maps). Vaggelis has not constructed them yet for KS, so we have no experience with them so far.  --- //[[predrag.cvitanovic@physics.gatech.edu|Predrag Cvitanovic]] 2009-04-14 06:24//


Suggested reading:

   * [[gtspring2009:gibson:w03#poincare_section|Gibson blog on poincare sections]]. 
   * [[http://www.chaosbook.org/chapters/cycles.pdf|ChaosBook.org/chapters/cycles.pdf]]

===== Factoring out the 4th-order translation symmetry =====
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