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Yes. every plot of a chaotic system is grossly wrong. But the cool thing is that it is imperceptibly different from another plot that is correct. this is a co
by ted_dunning 5y ago
Yes. every plot of a chaotic system is grossly wrong.
But the cool thing is that it is imperceptibly different from another plot that is correct.
this is a consequence of the shadowing theorem that says that while any finitely computed sequence has unavoidable errors, there is a sequence arbitrarily close to the numbers you do compute. (don't hold me to rigorous statements here, it has been many years) (the gist is right)
- PaulHoule 5y agoMy understanding is the shadowing lemma is controversial. For instance it applies to the chaotic orbits but not the stable orbits that are embedded in them.
- whatshisface 5y agoSince your finite-precision initial conditions are probably not on any stable orbits, I'm not sure how much that interferes with the truth of Ted's explanation.
- antognini 5y agoWouldn't the KAM theorem apply to the stable orbits? But for the chaotic orbits there was some good work a few years back by Boekholt & Portegies Zwart. They built an arbitrary-precision N-body integrator and found that the results of ordinary N-body simulations are statistically identical to arbitrary precision integrations: https://arxiv.org/abs/1411.6671 https://arxiv.org/abs/1411.6671