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--- gtspring2009:gibson:symbolic [2010/02/03 10:31]
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+++ gtspring2009:gibson:symbolic [2010/03/02 04:59] (current)
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 **2001-01-07** Fo' shizzle. I have one {e, sigma_x tau_xz} a.k.a. s2 symmetric orbit of period //T=113.8// with a //z// phase shift of //0.014 Lz//. That is slow: 1e-04 relative to the wall speed. One orbit does not a cycle expansion make, but at least I now know these are hard to find in their own right, rather than a mistake of mine.
 ====== orbits: what to do with them? ======
 **2010-02-03** I have some forty periodic orbits --a couple years' work but not enough for a cycle expansion //(quick, somebody send me an optimism pill).// The task at hand, then, is to understand how the existing orbits organize state space, and to see if we can bootstrap the generation of 10s of 1000s of orbits (and still not have enough). So I gotta find some interrelated orbits. Here's an interesting pair: P62p13 bifurcates off P30p99 at a lower Reynolds number. P30p99 has a τx phase shift, P62p13 does not.
-{{:gtspring2009:gibson:symbolic:2010-02-03-a.png?300}}
+{{:gtspring2009:gibson:symbolic:2010-02-03-a.png?300}}{{:gtspring2009:gibson:symbolic:2010-02-03-b.png?300}}{{:gtspring2009:gibson:symbolic:2010-02-03-c.png?300}}
+Almost all of the forty orbits are intersect the //I-D=0// section just once (oriented, and factoring out the discrete symmetries), and so are fixed points of the return map. The three orbits shown above (not counting P30p99) cross //I-D=0// multiple times and so are fixed points of the iterated map. P62p13 and P65p53 cross twice, and P76p77 three times. I suppose I will leave these orbits out of my first analysis of the return map fixed points, since technically they are not fixed points.
+ {{:gtspring2009:gibson:symbolic:2010-02-03-d.png?300}}
+L2 distances between fixed points of return map. The matrix shows the minimum of //|u_i - s_n u_j|// over the set of half-box shifts (under which the space of S symmetric fields is fixed). The blue spots tell us which orbits are near one another. I know that higher intelligences disdain energy norms (and that the distances computed here are not metrics for the //I-D=0// subspace), but since intelligent alternatives are not immediately available, we must make do as we creep our way towards them.
+**2010-02-04**
+Would like to bootstrap orbits but struggle to lift ideas from Smale horseshoe to plane Couette. Mayeb Siminos KS-RPO paper will help. //Later:// Hmmm, Predrag-Ruslan-Vaggelis also use close recurrences from simulation data as initial guesses for orbit searches.
+**2010-02-23 PC**
+//In celebration of the full day of work://
+that almost all of the forty orbits intersect the //I-D=0// section just once is probably expected. Think of it this way: in Predrag-Ruslan-Vaggelis KS paper one looks at the smallest cell in 1 dimension that is shmurbulent, and finds 3 equilibria and 2 relative equilibria, symmetry reduced. Shortest periodic orbits wind around these in various ways, so there might be 5.4/2 = 10 pairwise visitations which are the basic blocks from which all longer periodic orbits are glued together, and maybe show up as fixed points in the //I-D=0// section for KS. In plane Couette we have 3 dimensions, streamwise being the big deal, and spanwise and wall normal playing along. Each one could have of order of 10 important (relative) equilibria, and then the number of periodic orbits which are the shortest "building blocks" really shoots up. Now, because the box is minimal and motions along the 3 directions are strongly correlated, the number
+of basing building blocks is probably much smaller than this pessimistic estimate, but 40 is certainly not a large number.
+Hopefully physics will help us reduce them further - streamwise motions should be the most important ones, as this is how energy is fed into the flow, and spanwise much weaker. So symbolic dynamics might have triplets of symbols, one for each direction. It's going to be harder than KS. I wish Ruslan and Evangelos would return to slicing, because I see no other way of getting the symbolic dynamics for even 1D KS flow under control.
+**2010-03-01** This prompted me to do a calculation that's long been on my mind: see if any of the periodic orbits are winding around equilibria, as in the paradigmatic ODEs like Rossler and Lorenz. To do this I initiated a Newton-hookstep search an equilibrium solution on each of the orbits, using the I-D=0 intersections as starting points, on the reasoning that since eqbs satisfy I-D=0, those intersections are likely closest to any eqbs. The results were that each search took five to ten reasonably small steps downhill in residual, down to about 1e-04, but then got stuck at local minima. So that was 40 searches and no equilibrium solutions. Compare that to the 25/28 success rate in the smaller box for our 2009 JFM paper. Also, the Rossler and
+Lorenz dynamics are so controlled by those eqbs, I am sure that the same algorithm applied to most points in the natural measure would converge onto an eqb within a handful of steps. My interpretation is that these orbits are not governed by equilibria in the same way the Lorenz and Rossler are. To be more confident of this assertion I am starting searches on points spaced around the orbits at intervals of t=15 or 20.
+{{:gtspring2009:gibson:symbolic:2010-03-01-c.png?300}}
+**later same day** I did a check applying Newton-hookstep to the Lorenz equations using samples of integration as initial guesses for equilibria. The algorithm minimizes |f^T(x) - x| for fixed T. I was surprised to find how that strongly the success rate depended on T. For T ≈ 1/10 the oscillation time of the complex eigenvalue, the algorithm has about a 50% success rate. The above plot shows points that converged to equilibria as green dots (success) and those that got stuck in local minima in red (failure) for T=1 .
+{{:gtspring2009:gibson:symbolic:2010-03-01-d.png?300}}
+**Oops** I was misoverestimating the time scale of the oscillations. Hang on and I will revise. The oscillation time is about T=0.6, from the complex eigenvalue. (classic parameters sigma=10, beta = 8/3, rho=28). So a good small fraction of the oscillation time is T=0.1, and for that value, the success rate is 100%. Every initial guess I checked converges to one of the equilibria.

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