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gtspring2009:gibson:symbolic [2010/03/01 14:25]
gibson
gtspring2009:gibson:symbolic [2010/03/01 20:28]
gibson
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 {{:​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 for T on the order of a fraction of the complex oscillation. 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 strongly the success rate depended on T. For T << the oscillation time of the complex eigenvaluethe 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 .  ​
  
-**Oops** I was misunderestimating ​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%.+{{:​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.
  
  
gtspring2009/gibson/symbolic.txt · Last modified: 2010/03/02 04:59 by gibson