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It is a chaotic system. Arbitrary small deviations in the initial conditions will result in completely different outcomes. So your simulation will eventually di

It is a chaotic system. Arbitrary small deviations in the initial conditions will result in completely different outcomes. So your simulation will eventually diverge from reality as you cannot measure the initial conditions exactly.

zelphirkalt 5y ago
Nitpick + a little more thought: Isn't it more correct to say, that initially slightly different conditions might (instead of "will") result in a very much different outcome? Does chaotic mean, that two states which only differ a little must result in vastly different outcomes? I wonder whether there could be states, which are very similar and some condition drives them to converge again. Or is such a thing impossible?
- deleted 5y ago
  [deleted]
- jjgreen 5y ago
  Not to answer your question, but you may be interested to know that chaotic systems can often be effectively controlled by small perturbations: https://en.wikipedia.org/wiki/Control_of_chaos https://en.wikipedia.org/wiki/Control_of_chaos
- aaaaaaaaaaab 5y ago
  Indeed, there can be islands of stability in the phase space of chaotical systems.
- wisty 5y ago
  Often yes. How often happens in practice will usually depend on the size of the solution space. A chaotic system is pretty much a random number generator, and random number generators can spit out the same number (or nearby numbers) twice (otherwise they wouldn't be random).
- db48x 5y ago
  Yes, though frequently the solutions are very similar to each other. For example, if you plot the future of an asteroid in 10,000 different simulations, you’ll probably find that in most of them the asteroid remains in the asteroid belt where it started from, but perhaps in 10% of them it is perturbed enough by Jupiter that its orbit becomes a Trojan, or some other variety. If you look at the details of the 90% where it stays in the asteroid belt, you find that while they are all in different orbits from each other, the differences are not very significant. Just 9,000 rather similar orbits inside the asteroid belt. “Chaotic” usually means that the difference between two similar starting conditions grows without bound the longer you run the simulations forward. But orbits are closed loops; everything about an orbit is periodic. If two orbiting objects start near each other but have different orbital periods, then soon enough they will be far apart from each other. However, if you keep running time forward then they will end up right next to each other again. The distance between them is itself periodic, bounding the total error in a practical sense. Combine that with the overall stability of our solar system, and you find that most objects tend to stay in particular orbital families for quite some time. Most objects are near the bottoms of deep potential wells, and the forces that can push them out of those wells are quite small. It is only once they are pushed near the boundaries of those wells that rapid changes can begin to happen. Of course if it were any other way, then there would be nothing left in the asteroid belt by now. Compare that with Saturn’s rings, which simulations suggest will only last another 100k years, give or take a bit. They must be a relatively recent phenomena.
danwills 5y ago
I do accept what you have said (indeed it's chaos theory dogma that I am mostly still on board with!) but I also think reality is quite far from binary in whether chaos-theory concepts apply-to-it-or-not, seems a bit like it's mixed-in everywhere a bit, depending how information is mixing. (I'm calling chaos Class-3 Automata style) There are, I think still tools that we can build and use even in the face of this type of sensitivity to initial conditions!

zelphirkalt 5y ago

Nitpick + a little more thought: Isn't it more correct to say, that initially slightly different conditions might (instead of "will") result in a very much different outcome? Does chaotic mean, that two states which only differ a little must result in vastly different outcomes? I wonder whether there could be states, which are very similar and some condition drives them to converge again. Or is such a thing impossible?

deleted 5y ago
[deleted]
jjgreen 5y ago
Not to answer your question, but you may be interested to know that chaotic systems can often be effectively controlled by small perturbations: https://en.wikipedia.org/wiki/Control_of_chaos https://en.wikipedia.org/wiki/Control_of_chaos
aaaaaaaaaaab 5y ago
Indeed, there can be islands of stability in the phase space of chaotical systems.
wisty 5y ago
Often yes. How often happens in practice will usually depend on the size of the solution space. A chaotic system is pretty much a random number generator, and random number generators can spit out the same number (or nearby numbers) twice (otherwise they wouldn't be random).
db48x 5y ago
Yes, though frequently the solutions are very similar to each other. For example, if you plot the future of an asteroid in 10,000 different simulations, you’ll probably find that in most of them the asteroid remains in the asteroid belt where it started from, but perhaps in 10% of them it is perturbed enough by Jupiter that its orbit becomes a Trojan, or some other variety. If you look at the details of the 90% where it stays in the asteroid belt, you find that while they are all in different orbits from each other, the differences are not very significant. Just 9,000 rather similar orbits inside the asteroid belt. “Chaotic” usually means that the difference between two similar starting conditions grows without bound the longer you run the simulations forward. But orbits are closed loops; everything about an orbit is periodic. If two orbiting objects start near each other but have different orbital periods, then soon enough they will be far apart from each other. However, if you keep running time forward then they will end up right next to each other again. The distance between them is itself periodic, bounding the total error in a practical sense. Combine that with the overall stability of our solar system, and you find that most objects tend to stay in particular orbital families for quite some time. Most objects are near the bottoms of deep potential wells, and the forces that can push them out of those wells are quite small. It is only once they are pushed near the boundaries of those wells that rapid changes can begin to happen. Of course if it were any other way, then there would be nothing left in the asteroid belt by now. Compare that with Saturn’s rings, which simulations suggest will only last another 100k years, give or take a bit. They must be a relatively recent phenomena.

danwills 5y ago

I do accept what you have said (indeed it's chaos theory dogma that I am mostly still on board with!) but I also think reality is quite far from binary in whether chaos-theory concepts apply-to-it-or-not, seems a bit like it's mixed-in everywhere a bit, depending how information is mixing. (I'm calling chaos Class-3 Automata style) There are, I think still tools that we can build and use even in the face of this type of sensitivity to initial conditions!