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--- chaosbook:pipes [2010/09/16 09:43]
ashley
+++ chaosbook:pipes [2012/02/28 19:23] (current)
predrag
@@ Line 8: / Line 8: @@
 ====== A pipes blog ======
-**2009-08-24 On cylindrically symmetric codes** **[[http://www.math.lsa.umich.edu/~divakar/|Divakar Viswanath begin_of_the_skype_highlighting     end_of_the_skype_highlighting]]** talking to Predrag, about [[http://www.flickr.com/photos/birdtracks/3837531069/|Nicholas Grisouard]] and [[http://www.cims.nyu.edu/~obuhler/Oliver_Buhler/|Oliver Bühler]] [[http://nicolas.grisouard.free.fr/|code for "Bose-Einstein condensate"]] on a 2-dimensional disk:
+**2009-08-24 On cylindrically symmetric codes** **[[http://www.math.lsa.umich.edu/~divakar/|Divakar Viswanath]]** talking to Predrag, about [[http://www.flickr.com/photos/birdtracks/3837531069/|Nicholas Grisouard]] and [[http://www.cims.nyu.edu/~obuhler/Oliver_Buhler/|Oliver Bühler]] [[http://nicolas.grisouard.free.fr/|code for "Bose-Einstein condensate"]] on a 2-dimensional disk:
-  * Bessel ⇒ Hankel transform ⇒ inverse Hankel transform is a complicated code. The main expense is in quadratures. ONe also needs to precompute and tabulate the inverses. I could show Grisouard how to speed it up some.
+  * Bessel ⇒ Hankel transform ⇒ inverse Hankel transform is a complicated code. The main expense is in quadratures. One also needs to precompute and tabulate the inverses. I could show Grisouard how to speed it up some.
   * I follow [[http://www.comlab.ox.ac.uk/nick.trefethen/spectral.html|Nick Trefethen, Spectral Methods in MATLAB]], Chapter 11 and use Chebyshev polynomials. So does Rich Kerswell. Fourier and Chebyshev quadratures are essentially for free. Chebyshevs could be numerically unstable if too many points are used radially (200 or more), but so far no problem - in pipe simulations we use 10-20 radial point
@@ 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 129: / Line 129: @@
 **2010-09-16 Ash 2 Humble(dt)** I'm a habitual sinner - try clicking on the image now.
+**2010-11-23 Ash**  Been looking for recurrences in the turbulent cloud (entry 2010-07-08).  Its somewhat hampered by lots of jumps in the z-shift, //S_z//.  In Fourier space the tangent to //x// is generated by //t_a=Tx//, where //T=diag{im}//.  I've experimented with altering the definition just for the reference tangent //t'=diag{i.m^n}x'//.  More negative //n// helps but isn't great (see plot).  I've had a look at the equations for //dS_z/dt// but haven't thought of a reference tangent which that could avoid the jumps (i.e. avoids //g t_a . t'=0//).  Have there been any developments for the KS case?  Marc, any suggestions from your recent simulations for where this search may be fruitful?
+{{:chaosbook:pipes:2206_thetadot.png?400}}
+**2010-11-23 Humble(dt) 2 Ashley** I think Stefan Froehlich and I have now a clean presentation of how slicing works and precise statement how the reduced flow jumps through these singularities, but this is currently on siminos CNS subversion repository. If you guys have intellectual surplus to follow the latest slice & dice breakthroughs, I will have to make you jump a few firewall hoops. Have done it with Marc, but I suspect he has given up on reading these internal blogs. OK, telegraphic version: slicing is the only game in town, and will work, but you will have to construct multiple charts (slices centered on qualitatively different neighborhoods in the state space).
+**2010-11-24 Ash 2 Humble(dt)** I think I can see why there are likely to be jumps; by jumping through these, are you suggesting there's a particular slice it should jump to?  I'll have a go at tracking a few slices at once...
+**2010-11-30 Humbledt 2 Ashes** Try this - take two slice fixing points ('reference states'), each for a qualitatively different but  frequently visited neighborhood (a Mother and a Nigel?). The two slices are hyperplanes, so their intersection is also a hyperplane of a lower dimension. Swithc from slice to slice whenever you cross that hyperplane. We have not implemented this yet for Kuramoto-Sivashinsky, so I might be underestimating the difficulty of implementation, but hopefully it is a simple linear programming test, and you do not have to be precise about when you switch the slices - if we are lucky, all singularities are someplace further out on each slice that the boundary between them.
+**2011-01-12 Humbledt 2 plumbers** Dear Masters of Pipes, I've labored incredibly long to write a short intro to slicing for  Ashley and whoever has the courage to try it, and here is a readable draft,
+[[http://www.cns.gatech.edu/%7Epredrag/papers/preprints.html#FrCv11|"Reduction of continuous symmetries of chaotic flows by the method of slices"]] by Stefan Froehlich and Predrag Cvitanović. I tried to make it a few pages, but failed - would be grateful to anyone who tells me what more to cut/move to appendices, etc.
+Anyway, it's my firm belief that we can do it, so let's get to it.
+**2011-01-17 Ash** Following the entry on 2010-11-30, I've tried using all our TWs (for m=2, L=2.5D) as reference states and keep track the z-shift for each, //S_i(t), i=1..7// (1 Laminar; 2 LB; 3 M1; 4 M2; 5 UB; 6 S2a; 7 S2b).  For the plot below I start with index //iref=3//, and when //S_iref// crosses another //S_i//, I switch //iref// if we're closer to state //i//.  There are certainly fewer jumps in //dS_iref/dt//.
+{{:chaosbook:pipes:2210switching.png?400}}
+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 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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