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On Jan 1, 2008, at 8:19 PM, Ludwik Kowalski wrote:
1) What I would like to learn, from messages on this thread, is
how to distinguish a stable system from an unstable system.
Don't we need a definition of stability (for the kind of systems
being discussed)? If so then what is the acceptable definition?
2) This question is practical, not scholastic. The simulation
software I am using , I.P., seems to be highly reliable (consistent
with underlying physics). How can such software be used to
account for unavoidable perturbations? The I.P. does not account
for them. Please help me with this issue.
John Denker responded:
"Highly reliable"? That statement would be more
informative if it were more specific and more quantitative:
-- What tests have been run?
-- Quantitatively, how good was the agreement with
analytical results?
-- Are these tests designed to be incisive? What classes
of bugs are they likely to detect? Are they appropriate
to the numerical methods IP is actually using?
-- What are the /limits/ of validity?
-- How do we know that IP did not simply incorporate the
analytic solution for simple cases? (That's what I would
have done.) Doesn't that mean that the results for non-
simple cases will be incomparably less accurate?
How can such software be used to account for unavoidable
perturbations? The I.P. does not account for them.
Sure it can account for them.
*) For starters, you can run an ensemble of simulations, with
slightly different initial conditions.
*) Also you can perturb a two-body problem by adding a relatively
small third body.
*) Et cetera.