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--- chaosbook:pipes [2011/01/17 17:03]
predrag my answers to Ashley's comments on the Slice & dice paper
+++ chaosbook:pipes [2012/02/28 19:23] (current)
predrag
@@ Line 87: / Line 87: @@
 There are four N2 states at this wavelength.  Using the leading complex eigenvalue for the lower middle state (N2_M1) the corresponding eigenvectors have been used for projection with in the slice. N2_M1 has also been used for the reference to define the slice, it fills all coeffs within the subspace.
-{{:chaosbook:pipes:m1proj_full.png|}}
+{{:chaosbook:pipes:m1proj_full.png?400}}
 The lower state is on the boundary between the attraction to the laminar state (bottom right).  Not bad for a day's cycling.  Ash.
@@ Line 107: / Line 107: @@
 As we've already got states for L=2.5D, Re=2400, I think we should try opening up a symmetry first - it might not be necessary to change parameters yet...  Marc also suggests trying ICs in the other parts of the phase space, but keeping the current symmetries, in order to see if there's turbulence elsewhere.
-{{:chaosbook:pipes:rpo_n2srz2ub.png|}}
+{{:chaosbook:pipes:rpo_n2srz2ub.png?400}}
 **2010-06-04 Predrag** OK, stay here, open up the symmetry...
@@ Line 151: / Line 151: @@
 It picks up an excursion towards the LB state quite nicely, and for turbulence we find it hanging around S2a (see Kerswell & Tutty 07), but I'm still struggling to find orbits.  How should I pick the shift of the templates?  Having read Predrag's draft, maybe I should repeat adding, say, 100 randomly chosen turbulent states as templates?
-**2011-01-17 Humbledt 2 plumbers** No - as few slices as possible. And there should be no jumps in //dS_z/dt//, none at all. We just need to make sure that the ridges between them are sufficiently close to each the templates so that the inflection hyperplanes are excluded. Once templates are picked, the rest is geometry of hyperplanes (NOTHING to do with dynamics, only with the group theory) so checking whether the inflection hyperplane is on the far side of the tile edge (ridge between two slices) is a linear computation, to be undertaken independently of dynamics. I hope...
+**2011-01-17 Humbledt 2 plumbers** No - as few slices as possible. And there should be no jumps in //dS_z/dt//, none at all. We just need to make sure that the ridges between the templates are sufficiently close to each the templates, so that the inflection hyperplanes are excluded. Once templates are picked, the rest is geometry of hyperplanes (NOTHING to do with dynamics, only with the group theory) so checking whether the inflection hyperplane is on the far side of the tile edge (ridge between two slices) is a linear computation, to be undertaken independently of dynamics. I hope...
 For my answers to Ashley's comments on the Slice & dice paper, click {{:chaosbook:pc2aw.pdf|here}} (the problem is that if one is writing LaTeX, it's much easier to use siminos subversion blog, rather than copying stuff to here - dokuwiki is clunky when it comes to LaTeX.
+Regarding the latest figures - I usually think that time series in dynamics are not too insightful, would be nice to also see the reduced state space projections of the flow on one slice (for now), with segments of trajectory where //dS_z/dt// is large color-coded. Might see more clearly where the inflection hyperplane lies. I am bit worried about the shift velocities that you do see - I would hope there is a closer template in each case. so the singularity is preempted by switching to its neighborhood.
+**2011-02-16 Ash** Using the set of TWs as templates, I've been able to get quite long tracks without jumps.  This has meant that I could do some recurrence checking... Here's another RPO found using the slicing! [pink], //T=45 R/(2U)=11.25 D/U//; the crosses are equally spaced in time.  (S&Rot symmetry has been removed.) The candidate for the long-period orbit I've found difficult to get to converge.. I'm trying multiple shooting at the mo'.
+{{:chaosbook:pipes:2210nd_02.png?400}}{{:chaosbook:pipes:2514m1proj.png?400}}
+**2011-02-16 Humbledt** you are an angel, but I cannot see what the left figure is. Mea culpa, no doubt. Join us for Marc's Webinar, or be square.
+**2011-02-17 Ash** Sorry I missed the meeting, give me a bit longer warning next time.  I've registered but not found it online - is it recorded anywhere?  **Re yesterday's entry:**  The figure on the LHS shows the relative distance within the slice between states separated by time //dt//.  The signature around //t=3100..3300// suggested that the trajectory shadowed an orbit four or five times.  When plotted, the candidate trajectory looked like it might just be spiralling away from the S2a state (green spot on RH figure), but it proved worthwhile trying with Newton.  The plotted orbit was located within half a dozen iterations.
+**2011-03-31 Ash** Second RPO added to figure on 2011-02-16, the longer one (blue).  //T=37.93 D/U//;  marks on trajectories are //2 D/U// apart.  The orbit wanders roughly equidistant from M1 (lower blue circle) and S2a (green spot), not just around S2a as this particular projection suggests.
+**2011-04-14 Humbledt** Wow! this one will require some thinking. In the spirit of Froehlich & Cvitanovic paper, we will have to add a Poincare section hyperplane through each template point, start looking into local segments of the unstable manifolds to start working out symbolic dynamics for RPOs and making sense of it all. Already the (blue) RPO is too convoluted to interpret without making a stab at symbolic dynamics of longer orbits.
+**2011-04-08 Ash** And another (orange).  Very similar to the first RPO (pink), but having compared energies and friction factors, they are not just different projections of the same orbit.
+**2011-05-29 Humbledt** Got [[http://theanke.posterous.com/excellent-anglo-eu-translation-guide-albertoa|this link]] form Mason Porter, it might be helpful to non-Brits in this collaboration.
+**2011-05-29 Humbledt** Starting this date, the blog has been moved to CNS svn repository `pipes'.

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