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--- gtspring2009:gibson:symbolic [2010/03/01 13:44]
gibson
+++ gtspring2009:gibson:symbolic [2010/03/02 04:59] (current)
gibson
@@ Line 140: / Line 140: @@
 {{: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. Was surprised to find only 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). The algorithm minimizes |f^T(x) - x| for fixed T.
+**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}}
+**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.

channelflow.org

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