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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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>> 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.
I still don't really comprehend why RK4 is different to just integrating
in smaller steps. (That's all the algorithm appears to do.) But hey,
whatever.
> 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.
I've removed the singularities. They make the thing wildly unstable no
matter what integration method you use.
Even so, with Euler I couldn't make it stable, no matter how tiny I set
the integration steps. (I mean, I was approaching the limit of
machine-precision arithmetic with how tiny the steps were.) RK4 manages
apparently total stability with really quite large integration steps,
which is puzzling.
> There are much more robust methods, though:
>
> http://tableau.stanford.edu/~mwest/group/Variational_Integrators
> http://tableau.stanford.edu/~mwest/full_text/LeMaOrWe2004.pdf
>
> * http://www.scholarpedia.org/article/Runge-Kutta_methods#Stability
Heh. If I understood any of that, maybe I'd agree with you. ;-)
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Naughty boy. You've been doing a little bit too much acid. ;-)
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Attachments:
Download 'frame0039.jpg' (53 KB)
Preview of image 'frame0039.jpg'
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Kevin Wampler 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.
According to Wikipedia (which is never wrong), a chaotic system must
possess three attributes:
1. Sensitive dependence on initial conditions.
2. Topologically mixing.
3. Its periodic orbits are dense.
I know the system has property #1. I believe it has property #2. I have
no idea WTF #3 even *means*.
> 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.
According to Wikipedia, the important thing is that the orbits have
"significantly different" behaviour. (And apparently what you define as
"significant" can affect what counts as chaos.)
> 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.
Well, maybe...
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Pop quiz: The image shows signs of instability. (Those raggid edges are
probably supposed to be perfect curves.) How do I solve this?
1. Decrease the maximum time step size?
2. Decrease the minimum time step size?
3. Decrease the maximum error bound?
Answers on a postcard. ;-)
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Now with 2x supersampling antialias - and a lot less JPEG compression...
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Preview of image 'frame0039.jpg'
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Invisible wrote:
> Pop quiz: The image shows signs of instability. (Those raggid edges are
> probably supposed to be perfect curves.) How do I solve this?
>
> 1. Decrease the maximum time step size?
> 2. Decrease the minimum time step size?
> 3. Decrease the maximum error bound?
>
> Answers on a postcard. ;-)
Well, #1 is unlikely to have an effect. The operative question, though,
is whether the minimum time step size is actually being reached or not.
I don't have an answer to that question, but I just reduced the maximum
error bound ten-fold, and suddenly the images seem significantly less
"noisy".
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...and now with max_error = 0.001
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Attachments:
Download 'frame0039.jpg' (91 KB)
Preview of image 'frame0039.jpg'
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