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--- gtspring2009:gibson:w03 [2009/04/06 11:03]
gibson
+++ gtspring2009:gibson:w03 [2009/04/27 08:01]
predrag suggested a desymmetrized plot of Poincare sections
@@ Line 214: / Line 214: @@
 -04-06: added characteristic multiplier for period T=58.7 for each eigenvalue. %%mult = exp(57.8 Re λ)%%
+===== 2009-04-10 Poincare sections 3 =====
+{{gtspring2009:gibson:eq2poincare_ubef2_1.png?300}}{{gtspring2009:gibson:eq2poincare_ubef2_2.png?300}}{{gtspring2009:gibson:eq2poincare_ubef2_3.png?300}}{{gtspring2009:gibson:eq2poincare_ubef2_4.png?300}}
+The figures show the unstable manifold of EQ2 (the Nagata upper branch) projected onto an orthogonal
+basis set spanning the leading three eigenfunctions in the S-symmetric subspace, whose eigenvalues
+are listed above. e1, e2 span the complex instability λ = 0.0325 + i 0.1070 (n=1,2 in the above table).
+e3 is the component of the real part of the most weakly contracting eigenfunction (n=3,4 above,
+eigenvalue λ = -0.0141 + i 0.0577) that is orthogonal to e1,e2. That sets an arbitrary choice of
+orientation within the plane of the n=3,4 oscillation; it would perhaps be better to make the third
+coordinate in the plots below be the norm of the projection in this plane, or even the distance from
+u(t) to the e1,e2 plane. I will try that later.
+The green trajectories are in the unstable manifold of EQ2, and the blue orbit is P47p81. The short, thick
+black lines are the real and imaginary parts of the n=1,2 eigenfunctions. The short think red line is the
+real part of the n=3,4 eigenfunction. The long thin black lines are intersections of the unstable manifold
+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.
+===== 2009-04-13 Poincare sections 4 =====
+{{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
+repeats one shown immediately above, the right plot is the same but showing only the
+intersections with the Poincare section. The discontinuities in the unstable manifold
+are due to factoring out the 4th-order discrete translation symmetry. For any given state
+u(t), we apply whichever half-cell translation τ in {1,  τ<sub>x</sub>, τ<sub>z</sub>, τ<sub>xz</sub>} puts τ u(t) in the first quadrant of the [[gtspring2009:gibson:w03#inctro|four-fold symmetric pictures above]], that is meets the conditions
+<latex> $ \begin{align*}
+(1 + \tau_{x} - \tau_z - \tau_{xz} u_{EQ2}, \; \tau u(t)) & \geq 0 \\
+(1 - \tau_{x} + \tau_z - \tau_{xz} u_{EQ2}, \; \tau u(t)) & \geq 0
+\end{align} $ </latex>
+{{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).
+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//

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