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gtspring2009:gibson:symbolic [2010/03/01 14:25]
gibson
gtspring2009:gibson:symbolic [2010/03/02 04:59]
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 << ​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 oscillationIt 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 ≈ 1/10 the oscillation time of the complex eigenvaluethe 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 .  ​
  
-**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%. Every initial guess I checked converges to one of the equilibria.
  
  
gtspring2009/gibson/symbolic.txt · Last modified: 2010/03/02 04:59 by gibson