gtspring2009:gibson:symbolic
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gtspring2009:gibson:symbolic [2010/02/23 13:07] predrag on "forty orbits intersect the //I-D=0// section just once" |
gtspring2009:gibson:symbolic [2010/03/02 04:59] (current) gibson |
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| 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. | 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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| + | **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. | ||
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| + | {{:gtspring2009:gibson:symbolic:2010-03-01-c.png?300}} | ||
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| + | **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 . | ||
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| + | {{:gtspring2009:gibson:symbolic:2010-03-01-d.png?300}} | ||
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| + | **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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gtspring2009/gibson/symbolic.1266959232.txt.gz · Last modified: 2010/02/23 13:07 by predrag
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