gtspring2009:gibson:w03
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gtspring2009:gibson:w03 [2009/04/13 09:08] gibson |
gtspring2009:gibson:w03 [2009/04/27 08:01] predrag suggested a desymmetrized plot of Poincare sections |
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| with Poincare sections placed at θ = 0, pi/4, pi/2, 3pi/4, etc. The θ = 0 section is marked with small | with Poincare sections placed at θ = 0, pi/4, pi/2, 3pi/4, etc. The θ = 0 section is marked with small | ||
| circles and the θ = pi/4 with dots. | circles and the θ = pi/4 with dots. | ||
| - | |||
| ===== 2009-04-13 Poincare sections 4 ===== | ===== 2009-04-13 Poincare sections 4 ===== | ||
| - | {{gtspring2009:gibson:eq2poincare_ubef2_3.png?300}}{{gtspring2009:gibson:eq2poincare_ubef2_section_zero.png?300}} | + | {{gtspring2009:gibson:eq2poincare_ubef2_3.png?400}}{{gtspring2009:gibson:eq2poincare_ubef2_section_zero.png?400}} |
| Here we have the poincare sections at θ=0 (black, and θ=π in green). The left plot | Here we have the poincare sections at θ=0 (black, and θ=π in green). The left plot | ||
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| \end{align} $ </latex> | \end{align} $ </latex> | ||
| - | {{gtspring2009:gibson:eq2poincare_ubef2_2.png?300}}{{gtspring2009:gibson:eq2poincare_ubef2_section_pi2.png?300}} | + | {{gtspring2009:gibson:eq2poincare_ubef2_2.png?400}}{{gtspring2009:gibson:eq2poincare_ubef2_section_pi2.png?400}} |
| Same as above but with the Poincare section at θ=π/2 (black) and θ=3π/2 (green). | Same as above but with the Poincare section at θ=π/2 (black) and θ=3π/2 (green). | ||
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| - | {{gtspring2009:gibson:eq2poincare_ubef2_section_zeroB.png?300}}{{gtspring2009:gibson:eq2poincare_ubef2_section_zeroC.png?300}} | + | {{gtspring2009:gibson:eq2poincare_ubef2_section_zeroB.png?400}}{{gtspring2009:gibson:eq2poincare_ubef2_section_zeroC.png?400}} |
| Ok, some Poincare sections at θ=0 orientation, with vertical axis (u, e3) on left and | Ok, some Poincare sections at θ=0 orientation, with vertical axis (u, e3) on left and | ||
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| unstable manifold and plot a return map. This involves some more arbitrary decisions, namely, when to continue increasing arclength and when to project back onto previous portions of the unstable manifold. // 2009-04-13 12:07 EST// | unstable manifold and plot a return map. This involves some more arbitrary decisions, namely, when to continue increasing arclength and when to project back onto previous portions of the unstable manifold. // 2009-04-13 12:07 EST// | ||
| + | |||
| + | {{gtspring2009:gibson:eq2poincare_ubef2_section_zeroD.png?400}} | ||
| + | |||
| + | The plot above shows a number of points on the unstable manifold and their iterates under the return map. | ||
| + | A triangle symbol maps into the next triangle out along the unstable manifold, etc. The symbols are, in sequence, | ||
| + | triangle, square, diamond, star, circle. The first iterates of the symbols are all clustered just to the right of | ||
| + | EQ2 (at the origin). One return later they are spread from the triangle at (0.03, 0.03) to the circle at (-0.01, 0.08). | ||
| + | The next return, they are all over the map, and it no longer makes sense to draw lines connecting them (though I have | ||
| + | done so to show you that I am insane). | ||
| + | |||
| + | This is bad news, I believe. In order for the approximate return map s^(n+1) = f(s^n) (where s is pseudo-arclength | ||
| + | along the unstable manifold) to produce good guesses for periodic orbits, it needs to intersect the identity. | ||
| + | has to intersect the identity. In this plot, that would correspond to some portion of the lower part of the fold | ||
| + | to be mapped into the upper part. But the symbols show us this doesn't happen. | ||
| + | |||
| + | Corrections? Suggestions? | ||
| + | |||
| + | //John Gibson 2009-04-13 14:52 EST// | ||
| + | |||
| + | ===== Desymmetrized Poincare sections ===== | ||
| + | |||
| + | Can you plot these sections in desymmetrized space of Gibson //et al?// | ||
| + | |||
| + | {{:gtspring2009:howto:gtspring2009:2008-spring-p7.png?800}} | ||
| + | |||
| + | It should be easier to visualize them without discontinuities... --- //[[predrag.cvitanovic@physics.gatech.edu|Predrag Cvitanovic]] 2009-04-27 07:53// | ||
gtspring2009/gibson/w03.txt · Last modified: 2010/02/02 07:55 (external edit)
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