gtspring2009:gibson:w03

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2009-04-06: added characteristic multiplier for period T=58.7 for each eigenvalue. %%mult = exp(57.8 Re λ)%% | 2009-04-06: added characteristic multiplier for period T=58.7 for each eigenvalue. %%mult = exp(57.8 Re λ)%% | ||

- | ===== 2009-04-10 Poincare sections III ===== | + | ===== 2009-04-10 Poincare sections 3 ===== |

- | {{gtspring2009:gibson:eq2poincare_ubef2_1.png?300}}{{gtspring2009:gibson:eq2poincare_ubef2_2.png?300}} | + | {{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}} |

- | {{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// | ||

+ | ===== 2009-04-27 PC ===== | ||

+ | |||

+ | 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// | ||

+ | ===== 2009-08-05 ===== | ||

+ | |||

+ | I think the problem is not with the visualization but with the rapidity of stretching of the EQ2 unstable manifold, demonstrated by the motion of the symbols along the Poincare section of EQ2 shown above. But I agree that the discontinuities are a distraction. I was plotting in eigenfunction-based coordinates to try to more clearly illustrate the relation of the stretching and folding of the EQ2 unstable manifold with its eigenfunctions. | ||

+ | |||

+ | However in the meantime I think I've gotten a better idea of where this is going, with the Poincare sections. I have refactored my poincare-section code over the last few weeks to be more flexible. My last approach ended up requiring some pretty horrific shell work to massage the Poincare section data into manageable form. Now I have a command-line utility that will integrate from one crossing to the next, with the fundamental-domain mappings happening when they need to, and saving incremental data along the way for plotting. I think I can modify Arnoldi and Newton-Krylov algorithms without too much trouble so that they operate on section-to-section mappings. That will enable me to investigate the unstable manifolds of EQ2 and P48p18 orbits in a more canonical manner (e.g. like the Henon map in chaosbook). | ||

+ | |||

+ | I will restart with plots in which there are no discontinuities. Plots are coming. | ||

+ | ===== 2009-08-06 ===== | ||

+ | |||

+ | For now I will try to make progress understanding the EQ2 and P47 unstable manifolds via Poincare sections in a fundamental domain defined by a1 > 0, a2 > 0, where these are the coordinates in EQ2-translation basis, rather than using the nonlinear coordinate transformation from a → (a^2 - r^2)/r^2 as suggested above. Let me get some confidence in my Poincare section technology, and then I will apply it with a nonlinear coordinate transformation later, if that still seems sensible. | ||

+ | |||

+ | {{:gtspring2009:gibson:w03:eq2transfundamental.png?300}} | ||

+ | |||

+ | Here's a familiar picture of trajectories in the EQ2 unstable manifold viewed in an EQ2-translational basis. The blue trajectories are straight DNS integrations, the red are using some new code that applies a translation operation to u(t) and the DNS whenever it hits a boundary, and then computes Poincare section crossings on the trajectory restricted to the fundamental domain. The bold lines show the P47.18 periodic orbit, global (blue) and fundamental domain (red). | ||

+ | |||

+ | {{:gtspring2009:gibson:w03:eq2e1intersect1.png?300}} {{:gtspring2009:gibson:w03:eq2e1intersect2.png?300}} {{:gtspring2009:gibson:w03:eq2e1intersect3.png?300}} | ||

+ | |||

+ | The plots above are now restricted to the fundamental domain. The e0,e1,e2,e3 axes are the EQ2-centric translational basis, with zero-based indices (which are better for me computationally than one-based). The dots are intersections of trajectories with a Poincare section defined by (u(t)-EQ2, e1) == 0, with that quantity decreasing as it passes through zero. These plots are only orientation for what's coming. Also, I should note that the discretization here is really low in order to speed up the computations during development and testing phases, 32 x 33 x 32 (or 24 x 33 x 24 Fourier-Chebyshev modes, when you take dealiasing into account). Coefficients of order 1e-4 are getting cut off. EQ2 and P48p18 converge just fine on this smaller grid, but, for example, the period of the orbit differs in the fourth digit. When I encounter differences with greater sensitivity to discretization, I will note them. | ||

+ | ===== 2009-08-11 ===== | ||

+ | |||

+ | Further steps towards analysis of poincare section cutting EQ2 and P47. Have refined the integrations in order to get better resolution on the poincare section of the EQ2 unstable manifold. Blue lines are trajectories in 2d unstable manifold of EQ2. Red line is P47 orbit. Purple line is intersection of EQ2 unstable manifold w poincare section. Blue dot is EQ2. Red dot is P47 intersecting Poincare section. Projections, fundamental domain described above. Next step is to calculate the eigenvalues and eigenfunctions of P47's fixed pt in the poincare map and commpute P47 unstable | ||

+ | manifold in poincare section. | ||

+ | |||

+ | {{:gtspring2009:gibson:w03:2009-08-11-a.png?300}} {{:gtspring2009:gibson:w03:2009-08-11-b.png?300}} {{:gtspring2009:gibson:w03:2009-08-11-c.png?300}} {{:gtspring2009:gibson:w03:2009-08-11-d.png?300}} | ||

+ | |||

+ | Later p.m. news: have successfully recomputed P47 as a fixed point of the Poincare section map. Next step, eigenvalues and eigenvectors of the map about P47. | ||

+ | ===== 2009-08-12 ===== | ||

+ | |||

+ | Have computed eigenvalues of Poincare map about P47. The leading eigenvalues compare reasonably well to the eigenvalues of the orbit unconstrained to the section, but there are a large number of marginal eigenvalues and I suspect something is wrong, possibly sensitivity of Arnoldi to the way that data is fixed to lie on the Poincare section numerically. There are also differences between the maps and the discretization of the current computation is lower. I will recompute the eigenvalues of Poincare map and orbit in both discretizations. | ||

+ | |||

+ | ^ Poincare map ^^ Orbit ^^ 123 ^ | ||

+ | ^ abs λ ^ arg λ ^ abs λ ^ argλ ^ ^ | ||

+ | |21.68 | |22.60 | | SSS | | ||

+ | |9.256 | |9.254 | | AAS | | ||

+ | |4.784 | π |4.795 | π | SAA | | ||

+ | |2.302 | π |2.301 | π | SSS | | ||

+ | |1.874 | |1.873 | | ASA | | ||

+ | |1.808 | 0.47 π | 1.810 | 0.47 π | SSS | | ||

+ | |1.808 |-0.47 π | 1.810 |-0.47 π | SSS | | ||

+ | |1.4618 | |1.462 | | AAS | | ||

+ | |1.065 | π |1.065 | π | SAA | | ||

+ | |1.012 | |1.006 | 1e-04 | | | ||

+ | |1.0003 | 1e-04 |1.006 | -1e-04 | | | ||

+ | |1.0003 |-1e-04 |1 | | | | ||

+ | |0.9999 | |0.8577 | 1 | | | ||

+ | |many | | | | | | ||

+ | |more | | | | | | ||

+ | |marginal | | | | | | ||

+ | |||

+ | **update 2009--08-14:** the orbit eigenvalues in the table are now at the same discretization as the poincare map. | ||

+ | |||

+ | PC: as to the next step: instead of computing the P47 unstable manifold in Poincaré section I suggest parametrizing the EQ2 unstable manifold section in terms of arclenght, and studying how far iterates of points on it get from it in one iteration. If we are lucky, backfolding is strong enough to enclose the P47 fixed point within the first fold. | ||

+ | |||

+ | JFG: Check out [[#poincare_sections_4]]. This has symbols marking successive iterations of points along the EQ2 unstable manifold. Motion outwards along unstable manifold is very fast, and folding is hard to understand in so many dimensions. I was trying there to make folding clearer by plotting in non-orthogonal eigenfunction coordinates (two coords for the plane of complex oscillation, third being the weakest contracting direction). I think we need to understand what constitutes folding before making a return map along the unstable manifold arclength. Separation along only contracting directions makes sense as a definition of a fold, but detecting that in practice I don't have much sense for right now. | ||

+ | ===== 2009-08-14 ===== | ||

+ | |||

+ | (right) An arclength return map (purple) along the unstable manifold of EQ2 within the Poincaré section shown above. The dashed black line is the identity. Note that this is strict arclength, not pseudo-arclength (in which "folded" parts of an unstable manifold are projected back onto prior parts of the manifold), since I have not yet detected a fold and am not sure one even exists. The colored (orange, brown, blue) dots show successive iterates along the unstable manifold: for example, an orange dot maps into the next-furthest-out orange dot under one iteration of the poincare map. (left) the same dots shown on the state-space portrait of the Poincaré section. | ||

+ | |||

+ | {{:gtspring2009:gibson:w03:2009-08-14-a.png?400}}{{:gtspring2009:gibson:w03:2009-08-14-b.png?400}} | ||

+ | |||

+ | Comments: The initial slope of the return map is 6.43, roughly the same as the expansion factor 6.74 predicted by the complex eigenvalue of EQ2. The initial magnitude of the perturbation is fairly large (1e-03), so the discrepancy does not worry me. | ||

+ | |||

+ | PC: I think there is no need to actually compute the eigenvalues from the numerically computed Poincaré section, but it is comforting to check that the leading ones are in the right ball park. We know that they are the same as for the flow except for the one marginal along the time-evolution direction, which is eliminated by the section. The corresponding eigenfunctions for the expanding directions can be obtained by projecting the full space ones onto the Poincaré section tangent space at the point of the section. | ||

+ | |||

+ | JFG: If there is any folding going on in this unstable manifold, it is not obvious to me. Next I think I'll make a plot of |u_n - u_m|, or maybe just |u - u'| for u' = f(u) to see if there is any hint of folding. I realize that it would be better to compute distance only along unstable directions, but I am pretty sure that we are sufficiently far from the equilibrium that contracting vs expanding directions are unknown, and even if we were, the decomposition into stable versus unstable directions is not a trivial matter. | ||

+ | |||

+ | PC: I am not hopeful about folding back in the unquotiented statespace - there one might have only forward short-flight maps, from neighborhood to neighborhood - but I hope that in the Z_2 quotiented representation there is a return map that captures your short periodic orbit bouncing between the unstable Upper Branch equilibrium and its symmetry translate. |u(t) - u(t')| plot in the (t,t') plane for pairs of points on the unstable manifold (where distance is the energy norm measured in the Z2 quotiented statespace) sounds like a good idea for detecting any significant backfolding of the unstable manifold. You also have some intuition where the unstable manifold bends from your published 2-d projections of the unstable manifodl and it's symmetric sister. | ||

+ | |||

+ | JFG: These plots have the discrete symmetry quotiented out. The fundamental domain is one quadrant of the full space, with trajectories mapped back into the fundamental domain by translation whenever they exit. I have to run at the moment but the description of the quotiented should be described in detail above. **update** The quotienting is described and illustrated in my [[gtspring2009:gibson:w03#2009-08-06| 2009-08-06 entry]] |

gtspring2009/gibson/w03.1239404867.txt.gz · Last modified: 2009/04/10 16:07 by gibson