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gtspring2009:spieker_blog:eigenvectors [2010/02/02 07:55] (current)
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 +====== 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
 +
 +
 +
  
gtspring2009/spieker_blog/eigenvectors.txt · Last modified: 2010/02/02 07:55 (external edit)