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> But yes, I'm pretty sure this system possesses such a property.
It might depend on how much damping you give it.
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scott wrote:
>> But yes, I'm pretty sure this system possesses such a property.
>
> It might depend on how much damping you give it.
Doubt it.
Consider a point exactly between two attractors. A particle at this
point experiences zero resultant force. Purturbing the point by any
finite amount to either side will make the resultant force non-zero.
This will cause a different path to be traced, regardless of how much
damping is applied.
In general, applying more damping makes the system *less* unstable, but
does not remove areas of chaotic behavious; it just makes them smaller.
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Invisible wrote:
>> Now, if you add more than one attractor.....
>
> ...it goes completely scatty! o_O
Twisted and tangled, baby! :-D
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Attachments:
Download 'test_drawmany.jpg' (48 KB)
Preview of image 'test_drawmany.jpg'
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As an aside, I tried to implement this on my laptop at the weekend, but
it was hopelessly unstable. Today it seems very stable indeed. I can
only assume this is to do with replacing Euler with 4th-order
Runge-Kutta. I'm surprised it makes quite this much of a difference
though...
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Invisible wrote:
> Consider a point exactly between two attractors. A particle at this
> point experiences zero resultant force. Purturbing the point by any
> finite amount to either side will make the resultant force non-zero.
> This will cause a different path to be traced, regardless of how much
> damping is applied.
>
> In general, applying more damping makes the system *less* unstable, but
> does not remove areas of chaotic behavious; it just makes them smaller.
I think you may have some misconceptions about what chaos is. First
off, sensitivity to initial conditions is a necessary but *not
sufficient* condition for chaotic behavior. Secondly your example of a
point between the two attractors only argues for this sensitivity at a
manifold of points, and many many non-chaotic systems have such a
feature (for example a simple pendulum).
Thirdly, although it's possible I'm wrong here, if you have *any*
dampening I don't think the system can be counted as chaotic because all
paths will eventually converge to a point. If the system would be
chaotic without dampening it's sort of a minor point since it can still
look a lot like chaos, but technically I think it's incorrect to call it
chaotic.
Finally, I'm not sure that your system is chaotic. For inverse-square
springs it's known as Euler's three-body problem and appears to have a
(rather complicated) analytic solution.
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Kevin Wampler wrote:
> Finally, I'm not sure that your system is chaotic. For inverse-square
> springs it's known as Euler's three-body problem and appears to have a
> (rather complicated) analytic solution.
This is for just two attractors of course with three or more I'd be
pretty surprised if it weren't chaotic (and I wouldn't be too surprised
to find out I'm wrong about the two attractor system).
Also, neat pictures! You should color-plot the basins of attraction
when you have a dampening factor.
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Kevin Wampler wrote:
> Also, neat pictures! You should color-plot the basins of attraction
> when you have a dampening factor.
This is my goal. Unfortunately, the GTK+ subsystem is giving me some
spurious error message about a missing DLL or some such stupidity... *sigh*
--
http://blog.orphi.me.uk/
http://www.zazzle.com/MathematicalOrchid*
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On 30-4-2009 21:12, Orchid XP v8 wrote:
> Kevin Wampler wrote:
>
>> Also, neat pictures! You should color-plot the basins of attraction
>> when you have a dampening factor.
>
> This is my goal. Unfortunately, the GTK+ subsystem is giving me some
> spurious error message about a missing DLL or some such stupidity... *sigh*
>
You could use something like POV to do that for you...
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>>> Also, neat pictures! You should color-plot the basins of attraction
>>> when you have a dampening factor.
>>
>> This is my goal. Unfortunately, the GTK+ subsystem is giving me some
>> spurious error message about a missing DLL or some such stupidity...
>> *sigh*
>>
> You could use something like POV to do that for you...
I could use PPM too. It's just irritating that GTK+ has decided to stop
working today. :-/
--
http://blog.orphi.me.uk/
http://www.zazzle.com/MathematicalOrchid*
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On 30-4-2009 21:26, Orchid XP v8 wrote:
>>>> Also, neat pictures! You should color-plot the basins of attraction
>>>> when you have a dampening factor.
>>>
>>> This is my goal. Unfortunately, the GTK+ subsystem is giving me some
>>> spurious error message about a missing DLL or some such stupidity...
>>> *sigh*
>>>
>> You could use something like POV to do that for you...
>
> I could use PPM too. It's just irritating that GTK+ has decided to stop
> working today. :-/
>
I am sure you didn't change anything
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