Rob Freeman napsal(a):
> So chaos need not be all bad. It would make sense if we stored our
> information this way. Otherwise we would just be wasting bits.
Not to mention Ramsey's theorems from which it follows that complete
chaos is impossible :)
(just roughly: if you take a sufficiently large graph you will always
find a complete subgraph on n vertices or its complement - i.e. n
vertices with no edges between any two of them). (01)
> On Feb 6, 2008 2:18 AM, Ed Barkmeyer <edbark@xxxxxxxx> wrote:
>> Rob Freeman wrote:
>>> On Feb 5, 2008 3:54 AM, Ed Barkmeyer <edbark@xxxxxxxx> wrote:
>>>> Chaotic response is a behavior of a model under certain conditions. It
>>>> is proper to say that the model is probably not "valid" or "good enough"
>>>> under those conditions to predict anything about the system being
>>> Are you saying, not only "meaning" or the weather, but all chaotic
>>> behaviour is really only a manifestation of models which are not "good
>> No. I should have been more careful about what I wrote.
>> We have observed in many cases that mathematical models or discrete
>> simulation models of certain systems demonstrate chaotic responses to
>> certain stimuli, when the systems in question simply respond by moving
>> to a different, although less predictable, understood dynamic state. In
>> those cases, the problem is clearly that the model isn't faithful to the
>> behaviors of the system.
>> There are systems that actually have chaotic response to very minor
>> perturbations in certain regions, or in the self-symmetric case, in some
>> parts of every region.
>> My point was that one cannot conclude from chaotic behavior in a model
>> that that behavior is reflected in the behavior of the modeled system.
>> It may just be that the model breaks down in that region. We must come
>> to recognize true chaotic behavior by experimental observation.
>> And trying to validate that kind of stimulus-response performance
>> experimentally is quite tricky. You have to ensure that the values of
>> the two experimental stimuli are different but close enough to
>> demonstrate the chaotic response, and that usually requires very fine
>> control and measurement of the stimulus. Further, you have to be
>> convinced that there is no other uncontrolled variable operating to
>> produce the difference in effect. And most of the simulation model
>> failures occur precisely because they don't take account of some
>> unexpected influential variable. (The mathematical models tend to fail
>> because the model uses an understood function to approximate an unknown
>> one, and the behaviors of those functions diverge in some region. Once
>> you make reliable experimental observations in the region, it is easy to
>> see the functional divergence. That was the point of Mary Payne's
>> observation -- the tangent of the closest machine value to pi/2 is still
>> representable as a machine value, and the tangent of the next value
>> beyond that has a different sign.)
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