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 gtspring2009:schedule:sections_orbits [2009/04/10 08:10]gibson created gtspring2009:schedule:sections_orbits [2009/04/14 06:33]predrag 2009/04/14 06:33 predrag 2009/04/10 08:11 gibson 2009/04/10 08:10 gibson 2009/04/10 08:10 gibson created Next revision Previous revision 2009/04/14 06:33 predrag 2009/04/10 08:11 gibson 2009/04/10 08:10 gibson 2009/04/10 08:10 gibson created Line 1: Line 1: ====== April 14 study group ====== ====== 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 ===== ===== Poincare sections and periodic orbits ===== Line 15: Line 17: So the question is, how can we find approximate fixed points of the return map when there 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?​ 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// Line 20: Line 26: * [[gtspring2009:​gibson:​w03#​poincare_section|Gibson blog on poincare sections]]. ​ * [[gtspring2009:​gibson:​w03#​poincare_section|Gibson blog on poincare sections]]. ​ - * [[http://​www.chaosbook.org/​chapters/​cycles.pdf]] + * [[http://​www.chaosbook.org/​chapters/​cycles.pdf|ChaosBook.org/​chapters/​cycles.pdf]] ===== Factoring out the 4th-order translation symmetry ===== ===== Factoring out the 4th-order translation symmetry =====