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======= Symbolic dynamics in HKW cell =======

===== 2009-08-18 Drag = Dissipation Poincare section? =====

**JFG:** Here's the idea. Define a Poincare section by H = {u | h(u) = dE(u)/dt = I(u) - D(u) = 0 and dh/dt < 0}, I is wall shear and D is viscous dissipation. Energy balance requires that all periodic orbits intersect the section. So far all orbits intersect H transversely, and it seems unlikely that any will lie tangent to it. The intersections of the periodic orbits with the Poincare section form fixed points of the Poincare map. Label those fixed points a,b,c, etc. The Poincare section can be partitioned according to promixity to the fixed pts, e.g. B = {u in H | |u-b| < |u-a|, |u-b| < |u-c|, ...}. Then study symbolic dynamics of u. Implement control strategy to move about the Markov graph, etc.

We have 18 orbits in HKW right now, so that's at least 18 fixed points. I am currently calculating their I-D=0 intersections. 

Predrag, does that sound sensible?

===== 2009-08-22 =====

**PC:** Dunno. The intersections of the periodic orbits with the Poincare section are indeed periodic points of the Poincare map. We played with such ideas in [[http://www.cns.gatech.edu/~predrag/papers/preprints.html#rpo|our Kuramoto-Sivashinsky paper]]
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{{:gtspring2009:research_projects:gibson:rpopvd.png?700|}}
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Here every periodic orbit and relative periodic orbit has to cross the (P,D) diagonal, as its average is on the diagonal. My conclusion was that [power-in,dissipation,energy,...] plots are not illuminating, did not see how to get symbolic dynamics out of that. One of the reasons why working in the full state space seems the best strategy. [Of course, this paper is harder, because we are working in the full space, not a symmetry invariant subspace]. But it could be that it is an intelligent global Poincare section, and that if you look at the unstable manifolds of equilibria and periodic points (all equilibria are fixed points on this section) restricted to this section, and looked out in our full-state space Upper-Branch projections, you will indeed see nice foliation and an approximate symbolic partition of this (60000-1)-dimensional hypersurface. //I=D// is a curved hypersurface, not a hyperplane, right? 

===== 2009-08-24 =====

**JFG:** I'm not hopeful for looking at projections of unstable manifolds in this Poincare section, or looking at I vs D plots. For the former, the dimensionalities are still too large (as few as four and as many as twenty unstable directions per orbit, I think), and for the latter, we've done it and not learned much. What I do think I-D=0 Poincare section could get us is closer to understanding the network of transitions between the unstable periodic orbits. If, in the section, we form partitions according to the nearest fixed point, we can generate symbol sequences for that partition, and we could make a Markov model based on the symbol sequences. However I don't know a lot about this and so am asking a question about what can be accomplished with symbolic dynamics, if your starting point is a symbol sequence. The symbolic dynamics chapter in chaosbook (as I understand it, and I may be missing a lot) is about reducing several cases of known low-dimensional dynamics to unimodal 1d return maps. I think it's unlikely we'll be able to do that here. But I have a feeling that the symbol sequences from this partition will have some structure, and I hope that there are methods developed in the symbolic dynamics literature that will help us exploit that structure and in turn reveal something about turbulent dynamics. 

Yes, I=D will be a curved hypersurface in our energy-norm based projections, since D is a nonlinear function of u and I is linear. I am thinking of doing this with the HKW cell and the periodic orbits as fixed points of the Poincare map, rather than with the equilibria in the W03/GHC cell, because the Poincare section changes continuous time to discrete time and because we have many more orbits, all of which pierce the section, and all of which the dynamics visits, compared to the equilibria, most of which are 'out there'. 

**PC:** Your idea might bootstrap you to better and better symbolic dynamics in the following way:
  * determine all periodic points (points where your cycles pierce the D=I hypersurface; only the orbits with a single intersection yield fixed points) you have. The section usually has a sense of orientation, given by the local normal to it. If you allow for intersections of both orientations, each of your cycles has at least 2 periodic points, and probably the longer ones have n \approx T_p/t_min, where t_min is a typical shortest return time (there might be a number of different typical return times, for different regions of the section). What is good about your D=I hypersurface Poincare section is that 
    * it is physically motivated
    * all equilibria and relative equilibria lie on it, and their unstable manifolds can give you ideas about how to partition it
    * all periodic and relative periodic orbits must pierce it at least twice. There might be some non-generic tangencies, but that is OK
  * assign them neighborhoods/cells  by a Voronoy triangulation, ie a hyperplane through the shortest cord connecting each pair of neighboring periodic points
  * The periodic point goes to the next periodic point, with time being a fraction of the period of the orbit, so you know one directed link of the Markov diagram. Start at the edges of the cell to see what other cells you reach in short time. This gives you other links
  * use this Markov diagram to guess orbits of discrete (number of section crossings) time incremented by one
  * use segments of shorter periodic orbits that follow the same symbolic subsequence as starting guesses for Newton
this might produce a good hierarchy of longer and longer cycles. The longer segment you use, the better the guess, due to the exponential closeness of trajectories that share the same finite symbol sequence

===== 2009-08-25 =====


**JFG:** Good, I'm glad you think this is sensible. Most of the orbits I have so far pierce the I=D section just twice. Choosing the d(I-D)/dt < 0 orientation seems better to me in that these states are generally closer to the laminar flow: I-D = dEtot/dt  = 0 and d(I-D)/dt = d^2Etot/dt^2 < 0 gives a local maximum in total kinetic energy; but perturbation energy is generally lowest when total energy is highest (the laminar state has high total kinetic energy compared to a typical turbulent state, which has large velocity only near the walls). The partition I proposed is the same as Voronoy triangulation. It would be difficult to systematically explore the edges of the cells, as these are very high dimensional objects. Conceptually, one could examine only the intersections of the edges with the orbits' unstable manifolds with the edges, but this would still be quite difficult in practice. 

**PC:** The Voronoy section is high-dimensional, but it is pierced by the line connecting the two periodic points in a point - maybe one can use that point to initiate Newton guesses for one step longer orbits.

===== Ice fishing =====

**JFG 2009-10-01:** Ok, now I'm feeling the magic of Poincare sections. In the process of some other investigations I produced a long sequence of Poincare crossings {u_0, u_1, ...} of h = I-D = 0, dh/dt<0, for a single trajectory u(t). It then occurred to me to check these for close recurrences, i.e. small |u_n - tau u{n-k}| for small k and tau in the translation symmetry group under which S is invariant. One twenty-line program and half an hour later I have dozens of good initial guesses for periodic orbits, of periods ranging from T=25 to 300. This is not as efficient in terms of total integration time as is checking |u(t) - tau u(t-T)| over continuous t and T, but it is a hell of a lot easier, more organized, and amenable to doing systematically. It is still a total numbskull fishing approach, but perhaps in honor of the Poincare section and narrower range of scope we should call it "ice fishing."

**PC 2009-10-01:** Once, very long time ago, UFO and I were courted to move to Blackhole VA, and start an Institute of Non Science there. As a recruitment effort, the senior faculty member took me to his home's Entertainment Center, and there we watched (I swear, there is such a channel) The Fishing Channel, program on night fishing. For the most, the screen was black - then someone would show up under the light, clad in chest-high rubber pants, dangling a fish and grinning idiotically. Ever since I've been on lookout for thinking alternatives (thinking is extra price, but consider alternatives), lest Blackhole swallows me. 

**JFG 2009-10-01:** Fishing may be boring but it can be a good way to get fish. Between 10am and 2pm I found two new HKW orbits in four tries. I believe I'll be able to find dozens/hundreds of orbits this way without too much effort. I won't get burned at the stake for this, but it
s still worth doing, because it'll give us a better idea of how many orbits you need to build a crude dynamic model (linearizing about orbits, or even nearest orbit as zeroth order approx). From the perspective of fluid mechanics (my background and my best hope for future employment) this is important.
===== A weird symmetry =====

2009-11-29 While searching for relative periodic orbits in HKW with no symmetries, I found a solution //(u, σ, T)// where //σ : [u,v,w](x,y,z) → [-u,-v,w](-x + 0.626276... Lx, -y, z + 0.5 Lz)// and //T=30.99//. I had found an S-symmetric solution with the same period and a half-shift in x, so I compared the two and found, amazingly, that they were the same (identical velocity fields). That means the S-symmetric solution has the symmetry 

  [u,v,w](x,y,z) = [-u,-v,w](-x + 0.12627640259241246 Lx, -y, z + 0.5 Lz)

in addition to s1,s2,s3. I have checked the above numerically for u, and it is true to 1e-08. Small changes in the //x// shift disrupt the symmetry. This seems really weird to me, to have a phase shift that is not a simple fraction of the box size. It also makes me wonder if in our current set of solutions there are other strange symmetries which we haven't detected. The velocity field is pictured below.

{{:gtspring2009:gibson:hkw:p30p99.png?300}}

** 2009-11-30 PC**: Is this [[http://www.channelflow.org/dokuwiki/doku.php/database:hkw#p31.81|P31p81]]? How many digits of //0.12627640259241246// do you firmly believe? Eight? This number does not seem to be a rational or quadratic irrational - it has an unremarkable continued fraction expansion 

//[0; 7, 1, 11, 2, 1, 2, 1, 2, 4, 2, 3, 1, 1, 1, 5, 1, 9, 11, 1, 1, 1, 4, 1, 9]//

but who can tell from 8 digits. Sometimes there are symmetries under time reversal which are not spatial symmetries, and nontrivial algebraic relations among cycle points, but would not expect that for dissipative systems. Perhaps Γ/S factor/quotient group acts on cycle points, so //0.12627640 L<sub>x</sub>// is a point on the cycle //T/2// or //T/4// away, this prime cycle becomes periodic after quotienting. But would expect it to be a rational shift, and the period is already very short, though not much shorter than //P19p02//... Baffling.

** 2009-11-30 JFG** P30p99, P31p81, P31p17, and P41p36 are all connected via continuation [[:gtspring2009:gibson:continuation#section5| (see these figs)]]. I suppose that means all these solutions have the same weird symmetry. P41p36 is Kawahara and Kida's "gentle" solution. P30p99 is by far the most commonly found periodic orbit for this system. I trust the period and phase shift to 7 or 8 digits at the given spatial and temporal discretization, but only 3 or 4 digits in the //Nx,Ny,Nz,Δt → 0// sense. But I wouldn't think that finite spatial discretization would produce a non-simple-rational phase shift, given that the continuous symmetry is represented exactly at any discretization. Plus, on many occasions when searching for relative orbits and allowing continuous phase shifts, I find symmetric orbits with phase shifts of 1/2 and the 1/2 is accurate to 14 decimal places. At any given discretization, when I find the same orbit twice, the periods are equal to 8 or more digits. I will try finer resolution and see how much the weird phase shift changes.  Thanks for the continued fraction --what did you use to get it? Mathematica? 

** 2009-12-01 PC** [[http://www.wolframalpha.com/|wolframalpha.com]]

===== More weirdness =====

I am searching for relative periodic orbits // σ f^T(u) - u = 0// in the HKW cell using initial guesses //(u,σ,T)// found from close recurrences in successive crossings of the //I-D=0// Poincare section, either no symmetry or only //s1// or //s2// symmetry in //u(t)//, and with σ chosen by minimizing //|u^n - σ u^{n+1}|// over the //x// and/or //z// phase shifts as appropriate and the discrete symmetries. This produces a nice set of initial guesses, typically with nonzero guesses for the phase shifts. Typically the //x//phase shifts (when applicable) are distributed between  //[-Lx/2, Lx/2]// but the //z// phase shifts are usually within a few percent of zero, which is consistent with my observations that phase drift in //z// is very slow for this system. But, surprisingly, the vast majority of successful searches converge onto //s1,s2// symmetric periodic orbits with zero or half-box phase shifts! I know my code is working properly, since I found P30p99 as a relative periodic orbit with a weird phase shift, and I have checked it by recomputed one of the relative orbits that Divakar provided from a perturbed initial condition. So I am left to think that close recurrences in //u(t)// without symmetry restirctions are more commonly caused by close passes to symmetric periodic orbits than they are to nonsymmetric relative periodic orbits. This is not what we have expected.
===== Weirdness A and Weirdness B =====

I should be more careful about which discrete symmetry to add to the search space for the symmetry of relative orbits, since the discrete symmetries imply centers in //x// and/or //z//. I bet the weird phase shift I saw before was just s1,s2, or s3 in a shifted coordinate system, and of course I had to perform such a phase shift to make that solution fit the 0-centered definitions of s1,s2,s3. Will check and repost. 

**2009-12-01** Yep. The weird symmetry noted above is just //s2// under a particular phase shift. We should be able restate our 2009 JFM discussion of symmetry more powerfully. E.g. a symmetry involving a sign change in x,y and a non-zero phase shift in x is always the same as s2 under an appropriate x coordinate change, and then s2 itself implies periodicity at period Lx. We don't need to "restrict attention to half-box phase shifts", since any symmetry with sign change and shift is equivalent to half-box shift in a periodic box.

And for searching for solutions, the lesson is not to consider x,y sign-changing symmetry when searching for x-relative orbits. This is obvious in hindsight.

** 2009-12-01 PC** I'm also worried that we'll have eat our Halcrowiana. Not sure yet, but it seem that for pipes there are only 2 order two solution symmetries possible, so it's simpler than what we thought. Not sure yet.

** 2009-12-01 JFG** Pipe has just one discrete symmetry and plane Couette two because the nonzero downstream pressure gradient in pipes breaks disallows sign reversal in the downstream direction. That means the list of pipe symmetry groups is shorter and simpler than for plane Couette. No need to eat Halcrowiana, I think.

===== Periodic orbits vs Relative Periodic Orbits  =====

I figured out which discrete symmetries to include in the search for initial guesses for relative periodic orbits in s1-symmetric and s2-symmetric subspaces and over the last ten days have attempted perhaps twenty searches for relative periodic orbits in each of these subspaces: //s1//-symmetric orbits relative in //x//, and //s2//-symmetric orbits relative in //z//. Initial guesses were taken from close recurrences on the //I-D=0// section in //s1//-symmetric time series //u(t)// with the initial guess for //x// shift determined by minimizing //|u^n - tau u^{n+1}|//. Similarly for s2-symmetric searches.

I've found quite a few solutions but only one is a relative orbit. For example, for s2-symmetry I tried 19 searches for guesses with 5% initial error or less, and from these found 7 orbits, //all// of are true periodic orbits with s1,s2,s3. The initial guesses had  //s1 u ≠ u// but by the time the search finished, //s1 u = u//. Notably, the initial guesses for phase shifts in //z// are always quite small, order 0.01 Lz, which is compatible with observations that the spanwise drift of roll-streak structures is very slow. For s1-symmetric guesses, I have tried only 4 searches and have found one solution, which is a relative periodic orbit with symmetry 

  [u,v,w](x,y,z,88.5) = [u,v,-w](x+0.775Lx, y, -z, 0)

** 2009-12-01 PC** Spanwise drifts seems a weak effect, that is what you also noted when you were looking at `random walk/diffusion' of full space trajectories a couple of years ago - spanwise diffusion was so weak that it looked like ballistic motion for the time you run the simulation. Presumably because transfering energy from streamwise shearing to spanwise motion is a higher order, weak process. 

However, what about streamwise traveling waves and relative periodic orbits? They could have velocities of O(1) comparable to wall velocities. 

** 2009-12-21 JFG** The orbit noted just above is streamwise traveling (has an //x// shift). Over the weekend I found five more streamwise traveling relative periodic orbits. All these orbits have //s1// or equivalently //σz τxz// symmetry. The periods range from //T=90// to //T=110// and the phase shifts from //0.04 Lx// to //0.23 Lx//. The maximum possible phase shift is //0.5 Lx ≈ 3//, so no relative orbit will have O(1) phase speed unless //T// is O(1) as well. 
===== Nonlinear slicing =====

**2009-12-28 PC**, continuing on  //I=D//  Poincaré section discussion of 2009-08-25 above: 

//I=D// Poincaré section is a nonlinear hypersurface in the state space, //G//-invariant as it should be, pierced at least twice by all (relative) periodic orbits. It would be sweet if we could  use this physical idea to also construct a __nonlinear slice__ (slice is defined in [[http://chaosbook.org/paper.shtml#continuous|Sect. 10.4 0f ChaosBook]] - still being edited, click on the latest version before reading).

Do not have a clear idea (perhaps no idea at all?), but the hunch is something like this. D=I hypersurface Poincaré section condition is //dE/dt =0//. //E= <u,u>/2// is a G-invariant obtained by squaring and integrating over the equivariant velocity field //u//. We quotient //G// by slicing group orbits of //u// and encounter the problem that no single hyperplane slice can globally slice across all orbits. Can we think of a nonlinear slice manifold which, like the //dE/dt =0// condition cutting all periodic orbits, is guaranteed to cut across all group orbits?

**2010-01-04 JFG** That's a good question. 
===== Current orbit count =====

**2010-01-04** 35 periodic orbits in Rxz symmetry group, 15 relative orbits in {e, sigma_z tau_xz} with phase shift in x, none with z shifts. All the z-shift candidates converge onto fixed-z subspaces, so far. 

**2010-01-04** Whew/hallelujah! I finally seem to be converging onto an {e, sigma_x tau_xz} a.k.a. s2 symmetric orbit with a phase shift in z. Maybe several. It is all a matter of automation, and not having any mistakes in //Nx10^4// lines of code. Fear not, these are merely cooking in background while brain is applied to styling fine english cyclic prose and puffing up my ego in job applications.

**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-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.
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