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--- chaosbook:continuous [2010/06/06 01:45]
predrag SO(2) slicing discussion with Avitabile
+++ chaosbook:continuous [2010/06/06 05:53]
predrag Daniele Avitabile SO(2) x D_4 discussion
@@ Line 85: / Line 85: @@
 **2010-01-06 Predrag** For our PDE work I always use real 2D representation for Fourier modes, so we are on the same page. Group theory (Schur Lemma?) says that in defining an L2 product I am only allowed to use [2x2] identity multiplied by a constant on each such 2D linearly irreducible subspace, once for spanwise, once for streamwise, and no wall-normal symmetry reduction, other than what is included in discrete reflections. Give my best to the devil, tell him we got the message: if there is a symmetry, USE IT (long-winded version), or, JUST DO IT! (wisdom of Vaggelis T-shirts). If you check Siminos blog, for KS relative periodic orbits are starting to get reduced to periodic ones, and you can see what happens to slicing if higher Fourier coefficients of the slice point are too big in magnitude.
+===== A co-moving frame is not the answer =====
-======  Remarks ======
+**Predrag 2010-04-22 writes to Carles Simó** We've been arm-wrestling for a year now with continuous symmetries, more precisely how to go effectively to reduced state space M/G, where G is a continuous symmetry of dynamics (typically a compact Lie group such as O(2) X SO(2)) and M is extremely high-dimensional state space, such as a Navier--Stokes fluid). I think it would be simpler to do this first for dissipative, rather than Hamiltonian flows. Symmetry reduction has to be a post-processing method, as nobody is going to rewrite DNS code just to make me happy, and has to reduce a large set (for KS we have 40,000) of very different relative periodic orbits in M to periodic orbits in M/G. We have not find Marsdenise helpful. If you have a smart method how to do this, let me know?
+** Carles Simó** If you have something that seems to be a periodic orbit in some rotating frame, I would suggest to find the "rotation of the frame" (frequency analysis, e.g., can be helpful), undo the rotation at the end of the period and try to obtain the "clean" p.o.
+**Predrag** A co-moving frame is helpful in visualizing a single ‘relative’ orbit, but useless for viewing collections of orbits, as each one drifts with its own angular velocity. So we gave up on that early on.
+Visualization of all relative periodic orbits as periodic orbits we attain only by global symmetry reductions. We do it by "slicing", ie finding hypersurfaces that cut group orbits just like Poincare sections cut time-evolution orbits. The problem is that the natural choice;a hyperplane normal to a group orbit tangent vector (Lia algebra generator) is only good locally, globally it runs into singularities.
+===== Ginzburg-Landau system with SO(2) x D_4 symmetry =====
+**Predrag 2010-06-05** Calling an overall complex phase "gauge invariance" is a waste of an ugly but otherwise quite precise word. In quantum field theory one means by that the //local// gauge invariance (a very deep concept) in the original sense of Weyl, not the relatively trivial //global// gauge invariance. That is why in our papers and in the ChaosBook.org we never refer to "gauge fixing" when we discuss the choice of a slice, even though it is a method for fixing the phase of solutions. I'm saving the word for the tough, local "gauge invariance", if we ever get back from the classical PDEs to the quantum field theories. But AMO physicists do indeed use "gauge invariance" also for a global symmetry.
+**Daniele Avitabile  2010-06-05**
+Our problem has //SO(2) x D_4// symmetry. What we would like to achieve is exploring the symmetry-breaking bifurcation scenario pertaining to //D_4//, in the case of stationary solutions. In other words, we do not want to simulate the time-dependent solutions of the Ginzburg-Landau system. We concentrate instead on stationary solutions, and we want to see how, by breaking //D_4//, we can get new, hopefully observable, steady states.
+**Predrag** I think that you get equilibrium solutions only within subspaces fixed by //D_4// symmetries. The should be also //SO(2)// travelling wave solutions, perhaps also important. They would come in two flavors - with irrational velocities, and pre-periodic, belonging to a discrete //C_m// subgroup of //SO(2)//. And you can bifurcate into Hopf cycles, traveling waves, periodic orbits, relative periodic orbits etc., unless for some reason the physics does not like anything time-dependent.
+**Daniele** In order to do that we need to factor out the continuous group symmetry //SO(2)//. As far as I understood from your comments, we should be careful in doing that, as the the phase condition that we chose now  (seeking a solution in a hyperplane normal to a group-orbit tangent vector) is valid only locally. Am I correct in interpreting your statement?
+**Predrag** Yes. Though these singularities are perhaps easy to deal with...
+**Daniele** If so, I should say that we explore our stationary solutions only by means of numerical continuation. With this method, the best we can do *is to search locally*. What we need to make sure is that, once we have a solution and perturb it, we can still invert our problem and find a new steady solution. In other words, seeking for a solution locally is the best we can do, as far as I can tell. Do you think we could be missing some patterns by using our phase condition? I think your observation would be absolutely crucial if our continuous symmetry group was acting on the time-variable as well, and we wanted to simulate them. A typical case would be continuing time-dependent patterns like spiral waves or scroll waves. A method that comes to my mind in this context is the Freezing method by Tuemmler and Beyn, which you will probably know. I am pretty sure that this differs from the slicing method that you told us about. A recent article by Hermann and Gottwald recaps the method and contains an extensive set of references about it.
+**Predrag** I have read Tuemmler and Beyn and believe "freezing" = "slicing". Siminos and I have Hermann and Gottwald on our reading list, but have not studied it yet. Tuemmler and Beyn impose a slice by adding a new dimension (the phase parameter) and a condition (a Lagrange multiplier), while we go one dimension down by restricting the dynamics into the slice. Both approaches are standard in imposing Poincarė sections; Rytis Paskauskas have written that up for  [[http://chaosbook.org/paper.shtml#maps|Chapter 3 - Discrete time dynamics]] (not yet on the public version), and [[http://chaosbook.org/paper.shtml#cycles|Chapter 13 - Fixed points, and how to get them, Section 13.4 Flows]]. My feeling is that Tuemmler and Beyn is better, but we have not implemented it. If you do use it, let me know how it works.
+**Daniele** Now, let us assume that we manage to factor out //SO(2)//, either locally with our phase condition, or globally via the slicing method. You mention that you always quotient discrete symmetries. Would you recommend this in our case? What is the danger in not factoring them out? My gut feeling is that, by factoring //D_4//, for instance, we will make sure that we will find just one solution on the group orbit. I don't think this is bad in general, and I would be pleased to learn what are the problems associated with it.
+**Predrag** The quantum mechanics and the periodic orbit theory (see [[http://chaosbook.org/paper.shtml#symm|Chapter 21 - Discrete factorization]]) demands that discrete factorization be implemented - anything else would be stupid. Finding only 1 solution on a group orbit is not "bad", it's actually just great - and you can always return to the full state space by generating other solutions by the group actions.
+However, getting  one's students to do it is another story - the light goes on only after one has tried to use the periodic orbits one way or the other way (or tried to compute something in quantum mechanics with and without performing symmetry reduction first). People like to stick to full space, as discrete symmetry makes humanly "nicer" pictures, just like a kaleidoscope does.
+As far as I know, there is no price paid for reducing discrete dynamics (going from the full state space to a fundamental domain that tiles it) other than thinking.
+======  Remarks ======
 **2010-01-12 Dwight Barkley** Seems like a big pain to work through history. Would be easier if you waited until the participants were dead, don't you think? They can't complain then anyway.
 **2010-01-12 Predrag** Price one pays for being Teutonic. The problem is - who dies first? More of the problem is that I'm still crying out for help - I'm sure slicing can be done better for KS, spiral and fluid dynamics than what I have learned so far, so I'm hoping from enlightement from somebody from this wise crowd, so I'm trying to stay straight and narrow. If I do not mention them, they'll surely complain...
 ======  Literature ======
@@ Line 122: / Line 157: @@
   * A. Jadczyk, [[http://arkadiusz-jadczyk.org/papers/einstein.htm|Symmetry of Einstein-Yang-Mills systems and dimensional reduction]]. Would be lovely to get back to the that...  --- //[[predrag.cvitanovic@physics.gatech.edu|Predrag Cvitanovic]] 2009-12-17 23:17//
-**Predrag 2010-04-22 writes to Carles Simó** We've been arm-wrestling for a year now with continuous symmetries, more precisely how to go effectively to reduced state space M/G, where G is a continuous symmetry of dynamics (typically a compact Lie group such as O(2) X SO(2)) and M is extremely high-dimensional state space, such as a Navier--Stokes fluid). I think it would be simpler to do this first for dissipative, rather than Hamiltonian flows. Symmetry reduction has to be a post-processing method, as nobody is going to rewrite DNS code just to make me happy, and has to reduce a large set (for KS we have 40,000) of very different relative periodic orbits in M to periodic orbits in M/G. We have not find Marsdenise helpful. If you have a smart method how to do this, let me know?
-** Carles Simó** If you have something that seems to be a periodic orbit in some rotating frame, I would suggest to find the "rotation of the frame" (frequency analysis, e.g., can be helpful), undo the rotation at the end of the period and try to obtain the "clean" p.o.
-**Predrag** A co-moving frame is helpful in visualizing a single ‘relative’ orbit, but useless for viewing collections of orbits, as each one drifts with its own angular velocity. So we gave up on that early on.
-Visualization of all relative periodic orbits as periodic orbits we attain only by global symmetry reductions. We do it by "slicing", ie finding hypersurfaces that cut group orbits just like Poincare sections cut time-evolution orbits. The problem is that the natural choice;a hyperplane normal to a group orbit tangent vector (Lia algebra generator) is only good locally, globally it runs into singularities.
-**Daniele Avitabile  2010-06-05**
-Our problem has //SO(2) x D_4// symmetry. What we would like to achieve is exploring the symmetry-breaking bifurcation scenario pertaining to //D_4//, in the case of stationary solutions. In other words, we do not want to simulate the time-dependent solutions of the Ginzburg-Landau system. We concentrate instead on stationary solutions, and we want to see how, by breaking //D_4//, we can get new, hopefully observable, steady states.
-**Predrag** I think that you get equilibrium solutions only within subspaces fixed by //D_4// symmetries. The should be also //SO(2)// travelling wave solutions, perhaps also important. They would come in two flavors - with irrational velocities, and pre-periodic, belonging to a discrete //C_m// subgroup of //SO(2)//. And you can bifurcate into Hopf cycles, traveling waves, periodic orbits, relative periodic orbits etc., unless for some reason the physics does not like anything time-dependent.
-**Daniele** In order to do that we need to factor out the continuous group symmetry //SO(2)//. As far as I understood from your comments, we should be careful in doing that, as the the phase condition that we chose now  (seeking a solution in a hyperplane normal to a group-orbit tangent vector) is valid only locally. Am I correct in interpreting your statement?
-**Predrag** Yes. Though these singularities are perhaps easy to deal with...
-**Daniele** If so, I should say that we explore our stationary solutions only by means of numerical continuation. With this method, the best we can do *is to search locally*. What we need to make sure is that, once we have a solution and perturb it, we can still invert our problem and find a new steady solution. In other words, seeking for a solution locally is the best we can do, as far as I can tell. Do you think we could be missing some patterns by using our phase condition? I think your observation would be absolutely crucial if our continuous symmetry group was acting on the time-variable as well, and we wanted to simulate them. A typical case would be continuing time-dependent patterns like spiral waves or scroll waves. A method that comes to my mind in this context is the Freezing method by Tuemmler and Beyn, which you will probably know. I am pretty sure that this differs from the slicing method that you told us about. A recent article by Hermann and Gottwald recaps the method and contains an extensive set of references about it.
-**Predrag** I have read Tuemmler and Beyn and believe "freezing" = "slicing". Siminos and I have Hermann and Gottwald on our reading list, but have not studied it yet. Tuemmler and Beyn impose a slice by adding a new dimension (the phase parameter) and a condition (a Lagrange multiplier), while we go one dimension down by restricting the dynamics into the slice. Both approaches are standard in imposing Poincarė sections.
-Now, let us assume that we manage to factor out SO(2), either locally with our phase condition, or globally via the slicing method. You mention that you always quotient discrete symmetries. Would you recommend this in our case? What is the danger in not factoring them out? My gut feeling is that, by factoring D_4, for instance, we will make sure that we will find just one solution on the group orbit. I don't think this is bad in general, and I would be pleased to learn what are the problems associated with it.
 ~~DISCUSSION~~

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