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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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Invisible <voi### [at] dev null> wrote:
> 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...
The stability region for linear problems is a bit larger*, not to mention the
fact that it's, well, fourth order. Of course gravitation has ugly
singularities, so it usually just seems to be the case that it doesn't blow up
as badly as Euler near singularities. You can always divide by (r+epsilon)^2,
just to smooth things out--as long as you don't care about accuracy. There are
much more robust methods, though:
http://tableau.stanford.edu/~mwest/group/Variational_Integrators
http://tableau.stanford.edu/~mwest/full_text/LeMaOrWe2004.pdf
- Ricky
* http://www.scholarpedia.org/article/Runge-Kutta_methods#Stability
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Kevin Wampler <wam### [at] uwashingtonedu> wrote:
> 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.
Indeed, according to Richard Fitzpatrick:
http://farside.ph.utexas.edu/teaching/329/lectures/node57.html
In short, chaos requires
1. Aperiodic time-asymptotic behaviour
2. Deterministic
3. Sensitive dependence on initial conditions
> 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.
Transient != asymptotic. Good catch.
> 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.
I'm not sure that immediately disqualifies it. That's an interesting question
though. Can a dynamical system with an analytical solution be chaotic?
Certainly not without a series solution, but still. It's easy enough to come
up with aperiodic functions that solve a deterministic, dynamic system (sin(x)
+ sin(pi x), for one), but I can't think of any ODE's with an analytical and
chaotic solution.
- Ricky
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triple_r 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.
>
> I'm not sure that immediately disqualifies it. That's an interesting question
> though. Can a dynamical system with an analytical solution be chaotic?
> Certainly not without a series solution, but still. It's easy enough to come
> up with aperiodic functions that solve a deterministic, dynamic system (sin(x)
> + sin(pi x), for one), but I can't think of any ODE's with an analytical and
> chaotic solution.
Yeah, I wasn't sure either. On further investigation it looks like it
probably doesn't. I was able to find some references to chaotic
difference equations which had analytic solutions. I couldn't find
anything about ODEs though (at least not without wading through some
papers).
I'm now much more curious about whether or not the system is indeed
chaotic. If I get enough time to actually digest them, maybe I'll take
a deeper look at the analytic solution and see if I can figure out
what's going on there.
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