Are there such continuous chaotic systems, for which an explicit solution exists, which would allow to practically compute state at any given position in time just knowing the initial conditions? Or are they all the explicit solutions limited by the similar drawbacks as Sundman's solution of 3 body problem (extremely slow convergence of series in this case)?
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2Do you mean something like this: http://en.wikipedia.org/wiki/Logistic_map#Solution_in_some_cases – gammatester Feb 27 '14 at 14:35
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@gammatester This is an interesting example, thanks. But are there some continuous systems, like systems of differential equations, with such solutions? – Ruslan Feb 27 '14 at 15:24
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@Ruslan sorry for resurrecting this question after so much time, but do you have an English translation of Sundman's 1913 paper on the 3-body problem? – costrom Feb 08 '17 at 15:26
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@costrom no, I don't. Have there been any? – Ruslan Feb 08 '17 at 16:03
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@Ruslan not as far as I have found. I guess I'll have to do it then! – costrom Feb 08 '17 at 16:04
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The abstract of Ramahi and Seydou, A closed form solution of the Helmholtz equation for a class of chaotic resonators, Antennas and Propagation Society International Symposium, 2004. IEEE (Volume:4 ), says,
We study numerically the solution of the Helmholtz equation in a classically chaotic two-dimensional region in the shape of a bow-tie. The quantum ergodicity of classically chaotic systems has been studied extensively, both theoretically and experimentally, in mathematics and in physics. Despite this long tradition, we are able to present a new rigorous result using only elementary calculus. In particular, a closed form solution is derived by using multipole expansions. Our results have been validated by an integral equation method based on layer potential which is solved via the Nyström discretization method.
Gerry Myerson
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