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 gtspring2009:gibson:symbolic [2010/03/01 20:28]gibson gtspring2009:gibson:symbolic [2010/03/02 04:59] (current)gibson Both sides previous revision Previous revision 2010/03/02 04:59 gibson 2010/03/01 20:28 gibson 2010/03/01 14:25 gibson 2010/03/01 14:20 gibson 2010/03/01 13:44 gibson 2010/03/01 09:47 gibson 2010/02/23 13:07 predrag on "forty orbits intersect the //I-D=0// section just once"2010/02/04 15:49 gibson 2010/02/04 10:12 gibson 2010/02/03 13:35 gibson 2010/02/03 11:24 gibson 2010/02/03 11:23 gibson 2010/02/03 11:22 gibson 2010/02/03 10:31 gibson 2010/02/02 07:55 external edit 2010/03/02 04:59 gibson 2010/03/01 20:28 gibson 2010/03/01 14:25 gibson 2010/03/01 14:20 gibson 2010/03/01 13:44 gibson 2010/03/01 09:47 gibson 2010/02/23 13:07 predrag on "forty orbits intersect the //I-D=0// section just once"2010/02/04 15:49 gibson 2010/02/04 10:12 gibson 2010/02/03 13:35 gibson 2010/02/03 11:24 gibson 2010/02/03 11:23 gibson 2010/02/03 11:22 gibson 2010/02/03 10:31 gibson 2010/02/02 07:55 external edit Line 140: Line 140: {{:​gtspring2009:​gibson:​symbolic:​2010-03-01-c.png?​300}} {{:​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 strongly the success rate depended on T. For T << ​the oscillation time of the complex eigenvalue, the algorithm has about a 50% success rate. It 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 .  ​ + **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}} {{:​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%. Almost every initial guess on the natural measure ​converges to one of the equilibria. + **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.