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

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?

I'll try to upload later today the 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 2009-04-14 06:24

Suggested reading:

• Gibson blog on poincare sections.
• ChaosBook.org/chapters/cycles.pdf