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 — gtspring2009:spieker_blog:eigenvectors [2010/02/02 07:55] (current) Line 1: Line 1: + ====== Eigenvectors of the Upper Branch for chaosbook ====== + + [[:​gtspring2009:​spieker_blog:​UB_eigenvectors:​first_try|First Attempt]] + + The following links are provided in order to maximize the aesthetics and bring out salient features of the eigenvectors,​ so that PC may more easily gauge what would be perfect for chaosbook, kinda like looking at paint chips: + + [[:​gtspring2009:​spieker_blog:​UB_eigenvectors:​color_scaling|Color Scaling]] + + [[:​gtspring2009:​spieker_blog:​UB_eigenvectors:​laminar|Laminar?​]] + + ====== Frequency of occurrences in searches of existing orbits ====== ​ + + ** DWS starting 10/15/2009; ** + + **HKW** + + ^orbit^# of occurrences^ + | T=65.53 | 1 | + | T=75.35 | 1 | + | T=76.85 | 1 | + | T=87.89 | 1 | + + **W03** + + ^orbit^# of occurrences^ + | EQ14 | 2 | + | EQ15 | 2 | + + ====== Floquet exponents/​multipliers ====== ​ + + ** 10/16/09 DWS ** Working on the Floquet exponents/​multipliers of the new periodic orbits. ​ I wanted them to be in the correct size boxes before I calculated them. As far as the the equilibria go, here are their first 10 eigenvalues (maybe needs a page of its own for organizational reasons: + + **EQ14** + + ^ No. ^ Re(lambda) ^ Im(lambda) ^ abs(Lambda) ^ arg(Lambda) ^ + | 1 | 0.0253 | -0.00857 | 1.28885 | -0.0857 | + | 2 | 0.0253 | 0.00857 | 1.28885 | 0.0857 | + | 3 | 0.00847 | 0 | 1.08843 | 0 | + | 4 | 3.53e-6 | 0 | 1.00004 | 0 | + | 5 | -2.89e-7 | 0 | 1.00000 | 0 | + | 6 | -7.58e-3 | 0 | 0.927024 | 0 | + | 7 | -8.48e-3 | 0 | 0.918696 | 0 | + | 8 | -1.04e-2 | 0 | 0.901207 | 0 | + | 9 | -2.45e-2 | 0 | 0.783082 | 0 | + | 10 | -3.03e-2 | 0 | 0.738803 | 0 | + + **EQ15** + + ^ No. ^ Re(lambda) ^ Im(lambda) ^ abs(Lambda) ^ arg(Lambda) ^ + | 1 | 4.12e-2 | 0 | 1.50924 | 0 | + | 2 | 3.66e-2 | -3.58e-02 | 1.44230 |-0.3584| + | 3 | 3.66e-2 | 3.58e-02 | 1.44230 |0.3584| + | 4 | 6.79e-3 | 0 | 1.07024 | 0 | + | 5 | 3.04e-6 | 0 | 1.00000 | 0 | + | 6 | 4.74e-7 | 0 | 1.00000 | 0 | + | 7 | -7.47e-3 | -4.18e-3 | 0.928024 | -4.17168e-3 | + | 8 | -7.47e-3 | 4.18e-3 | 0.928024 | 4.17168e-3 | + | 9 | -1.82e-2 | -0.101 | 0.833538 ​ | -1.01328 | + | 10 | -1.82e-2 | 0.101 | 0.833538 | 1.01328 | + + ** P59.77 ** + + ^ No. ^ Re(lambda) ^ Im(lambda) ^ abs(Lambda) ^ arg(Lambda) ^ + | 1 | 2.109e-2 | -9.198e-3 | 3.52807 | -0.549753 | + | 2 | 2.109e-2 | 9.198e-3 | 3.52807 | 0.549753 | + | 3 | 1.700e-2 | -3.052e-2 | 2.76295 | -1.82413 | + | 4 | 1.700e-2 | 3.052e-2 | 2.76295 | 1.82413 | + | 5 | 3.317e-3 | 5.256e-2 | 1.21926 | 0 | + | 6 | 2.890e-3 | 0 | 1.18854 | -1.95659 | + | 7 | 3.734e-4 | -3.274e-2 | 1.02257 | 1.95659 | + | 8 | 3.734e-4 | 3.274e-2 | 1.02257 | 0 | + | 9 | 7.884e-5 | 0 | 1.00472 | 0 | + | 10 | 3.168e-6 | 0 | 1.00019 | 0 | + + ** 10/17/09 DWS ** I think the 5th imaginary value above should be zero as the arg(Λ) was π. + + ** 10/20/09 PC ** There is no point doing these tables in dokuwiki format. You need them in your thesis, and John and I have to agree on how to present periodic orbits in our periodic orbits paper, so I have toiled away for past two days on a proposal how to tabulate them. DWS, JFG, ES, and RLD, please check out ChaosBook.org section [[http://​chaosbook.org/​chapters/​appendStability.pdf|B.3 Eigenspectra:​ what to make out of them?]]. + + ** 10/21/09 DWS ** Okay, so after a few passes I think I understand everything you propose we calculate for the periodic orbits. ​ I also got my svn working again and I will put the tables in there for my thesis. ​ I am not 100% sure, but I believe that the eigenvectors produced by arnoldi iteration are orthogonal because I believe that arnoldi iteration, by definition, uses the stabilized Gram-Schmidt process. ​ I also checked a number of eigenvectors that I produced using the arnoldi utility and by using the L2IP function, I found that the inner product between two different eigenvectors was always very close to 0, usually within 10^-5. ​ I would like confirmation of this from John G., though. + + ====== Solution Eigenvalue Tables (for my own reference) ====== + FIXME + + EQ1 + + ^ n ^ Re(lambda) ^ Im(lambda) ^ s1 ^ s2 ^ s3 ^ + | 1 | 5.0120784E-2 | 0 | + | 2 | 2.2895976E-6 | 0 | + | 3 | -4.646974E-7 | 0 | + | 4 | -2.004801E-3 | 0 | + | 5 | -6.599051E-3 | 0 | + | 6 | -6.929019E-3 | 0 | + | 7 |  | 0 | + | 8 |  | 0 | + | 9 |  | 0 | + | 10 |  | 0 | + | 11 |  | 3.43223 | + + + EQ2 + + ^ n ^ Re(lambda) ^ Im(lambda) ^ s1 ^ s2 ^ s3 ^ + | 1 | | | | | | + EQ3 + + EQ4 + + EQ5 + + EQ6 + + EQ7 + + EQ8 + + +