gtspring2009:gibson:symbolic

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gtspring2009:gibson:symbolic [2010/03/01 20:28] gibson |
gtspring2009:gibson:symbolic [2010/03/02 04:59] (current) gibson |
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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 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 . |

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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%. 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. |

gtspring2009/gibson/symbolic.txt · Last modified: 2010/03/02 04:59 by gibson