gtspring2009:gibson:symbolic
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gtspring2009:gibson:symbolic [2010/03/01 09:47] gibson |
gtspring2009:gibson:symbolic [2010/03/02 04:59] (current) gibson |
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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 | **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. | 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.1267465649.txt.gz · Last modified: 2010/03/01 09:47 by gibson
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