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gtspring2009:gibson:w03 [2009/04/13 08:37]
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). ​
 +
 +The θ=0 section above looks potentially fruitful. It could be that the unstable manifold is
 +folding back on itself due to the desymmetrization. Or it could be an artifact of the projection due to the arbitrary choice of the third (vertical) axis in the plots, e3. I will plot a few other projections of the same section to see. // 2009-04-13 11:40 EST//
 +
 +
 +{{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
 +(u, e4) on right. e3, e4 are the orthogonal components of the real/​imaginary parts of
 +the n=3,4 eigenfunction,​ respectively. ​ I have marked a few points on the unstable manifold with symbols so they can be compared between plots. Next is to assign arclengths along
 +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)