====== 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