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All of the fields with and without the laminar field subtracted. Laminar subtracted on the left, laminar omitted on the right: {{:gtspring2009:spieker_blog:ub_eigenvectors:laminar:ef1_lam.png|Ef1 Laminar}} {{:gtspring2009:spieker_blog:ub_eigenvectors:laminar:ef1_nolam.png|Ef1 Non-laminar}} {{:gtspring2009:spieker_blog:ub_eigenvectors:laminar:ef2_lam.png|Ef2 Laminar}} {{:gtspring2009:spieker_blog:ub_eigenvectors:laminar:ef2_nolam.png|Ef2 Non-laminar}} {{:gtspring2009:spieker_blog:ub_eigenvectors:laminar:ef3_lam.png|Ef3 Laminar}} {{:gtspring2009:spieker_blog:ub_eigenvectors:laminar:ef3_nolam.png|Ef3 Non-laminar}} {{:gtspring2009:spieker_blog:ub_eigenvectors:laminar:ef4_lam.png|Ef4 Laminar}} {{:gtspring2009:spieker_blog:ub_eigenvectors:laminar:ef4_nolam.png|Ef4 Non-laminar}} {{:gtspring2009:spieker_blog:ub_eigenvectors:laminar:ef5_lam.png|Ef5 Laminar}} {{:gtspring2009:spieker_blog:ub_eigenvectors:laminar:ef5_nolam.png|Ef5 Non-laminar}} {{:gtspring2009:spieker_blog:ub_eigenvectors:laminar:ef6_lam.png|Ef6 Laminar}} {{:gtspring2009:spieker_blog:ub_eigenvectors:laminar:ef6_nolam.png|Ef6 Non-laminar}} I could go either way on whether or not to include the laminar solution on top of the eigenvalue. {{gtspring2009:pc.jpg}} Dustin or JohnG, can you explain why one would subtract laminar from eigenvectors? Adding a constant to //**u**(x,t)// does not change the matrix of state-space velocity gradients //A//, or its eigenvalues, eigenvectors, or am I missing something? --- //[[predrag.cvitanovic@physics.gatech.edu|Predrag Cvitanovic]] 2009-03-31 07:10// {{gtspring2009:pc.jpg}} Dustin, in [[gtspring2009:spieker_blog:ub_eigenvectors:first_try|your first try]] I went through the exercise of guessing which of your eigenvectors correspond to which eigenvalues. Did you recheck my labeling? It's important one gets this correctly to get the right behavior and right symmetries in close passages to equilibria, and in particular for their heteroclinic connections. Also, I do need the least contracting stable eigevalue (pair), JohnG and I believe that it offers a natural axis for local //3D// plots. --- //[[predrag.cvitanovic@physics.gatech.edu|Predrag Cvitanovic]] 2009-03-31 07:10// :-) I do believe that your labelling is correct because it corresponds to John's description of the full-space eigenvectors that he listed in his Poincare Section write-up. I have not performed the calculation restricted to the SSS subspace as John has and found the first 30 eigenvectors in that subspace. I am also unsure of the meaning of "2 marginal eigenvalues - do they have nontrivial eigenvectors?" I also am 90% certain that the pair of Re %%e%%_11, Im%%e%%_11 does form the least contracting eigenvalue pair. --- //[[dustin.spieker@gatech.edu|Dustin Spieker]] 2009-03-31 09:11// {{gtspring2009:pc.jpg}} Should you scale colors to bring some structure in the leading eigenvector **e**_1? In principle the most important one in the full state space? Also, Im **e**_2 is pale compared to Re **e**_2, but they should be roughly the same intensity, as they span a spiral-out plane. Seems like manually adjusting these colors is a pain; can you scan the picture for redest/bluest, and then use these for a full rescale up to max blue (or red, whichever is larger in the orignal version). --- //[[predrag.cvitanovic@physics.gatech.edu|Predrag Cvitanovic]] 2009-03-31 07:40// {{gtspring2009:pc.jpg}} I'm curious: what do the the eigenvectors of the two marginal eigenvalues look like? They should point along the two continuous translations, streamwise and spanwise. Perhaps obvious... --- //[[predrag.cvitanovic@physics.gatech.edu|Predrag Cvitanovic]] 2009-03-31 07:10// {{gtspring2009:pc.jpg}} responding to Dustin 2009-03-31 above: Thanks for confirming that list agrees with JohnG's. For each continuous symmetry there is one marginal eigenvalue whose value is exactly 0; I assume that the code generates two of those to at list 6 significant digits precision. If code alos generates two associated eigenvectors, I am curious what they look like. --- //[[predrag.cvitanovic@physics.gatech.edu|Predrag Cvitanovic]] 2009-04-01 02:54//
