Could anyone please tell me an example of an equation with no analytic solution(s) that is not a chaotic one? And what is the physical meaning of having analytic solution? For instance, the three body problem does not have in general analytic solution and it leads to chaos. But I don't know if this is a general statement. I have absolutely no idea. Could anyone explain me, please?
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$\begingroup$ What do you mean by analytic? Do you mean real/complex analytic or just expressible in terms of familiar functions? $\endgroup$– Ryan ThorngrenCommented Jan 31, 2021 at 22:26
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3 Answers
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Could anyone please tell me an example of an equation with no analytic solution(s) that is not a chaotic one ??
A simple fifth order polynomial ($k_5 x^5 + k_4 x^4 + k_3 x^3 + k_2 x^2 + k_1 x + k_0 = 0$) has no analytic solution, but is not chaotic.
And what is the physicall meaning of having analytic solution ??
There is no physical meaning. Nature doesn’t care if we have nice functions to describe something.
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$\begingroup$ Thanks a lot !! I can't give you a like because I'm not important enough . Furthermore I didn't think about Galois theorem. Although that equation sometimes has analytic solutions indeed. Everytime it can be decomposed in a direct product of groups. And how do you know whether or not a polynomial equation leads to chaos ?? $\endgroup$– EvaristeCommented Jan 31, 2021 at 22:50
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1$\begingroup$ I don’t think that there is a general test for chaos. I think you just have to look at the solutions. An ordinary fifth order polynomial just has 0 to 5 real roots, and as a function no more than 4 extrema, so nothing chaotic. $\endgroup$– DaleCommented Jan 31, 2021 at 23:24
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$\begingroup$ I don't believe this is correct. Galois theory tells us that there's no general solution to the general quintic in radicals, but it certainly can be solved analytically using elliptic functions. Sextics require more general functions but I believe it has been proved that there are families of analytic functions that solve polynomials to all orders. $\endgroup$ Commented May 15 at 4:46
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As a real everyday physical model, consider a non-damping pendulum: ($\ddot{\theta} + \frac{g}{l}\sin\theta = 0$). It does not have a general analytical solution (Wikipedia) and yet it is not chaotic.
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$\begingroup$ Thanks to you as well !! People are fantastic here !! : D $\endgroup$– EvaristeCommented Feb 6, 2021 at 1:55
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$\begingroup$ This is not correct either I'm afraid. Theta's time dependence is just Jacobi's amplitude, which is very much analytic. The x, y coordinates are then given by the sinus and cosinus amplitudinus. These functions have a great theory and can be evaluated very efficiently. You do not get chaos, because everything is smooth I'm afraid. $\endgroup$ Commented May 15 at 4:48
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Like the following?
$$ \sin (x) = \lambda x \tag{1} $$ for $\lambda < 1$
Or do you want an ODE? You did not specify in the question.
answered Jan 31, 2021 at 23:59
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$\begingroup$ Thanks to you too. There are plenty of them !! I was figuring out a kind of relation between those two concepts, but now I realize that such a relation does not make any sense. $\endgroup$– EvaristeCommented Feb 1, 2021 at 7:46
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$\begingroup$ Even though the above has smooth solutions (not analytically described) it can still be used to generate chaos using the recursion $$x \leftarrow \frac{\sin(x)}{\lambda}$$ $\endgroup$ Commented Feb 1, 2021 at 13:28