Wikipedia:Reference desk/Archives/Mathematics/2015 December 10

= December 10 =

Chaotic systems with closed-form solutions
I know that in finite dimensions, a chaotic system has to be nonlinear, but are there any that have closed-form solutions? It seems like all the examples in the articles have to be solved numerically.--Jasper Deng (talk) 04:02, 10 December 2015 (UTC)
 * Iterating $f(x) = 4x(1−x)$ is chaotic and has the closed-form solution $f ^{n}(x) = sin^{2}(2^{n} arcsin(√x))$, as stated in Iterated_function and Schröder's_equation. Egnau (talk) 15:27, 10 December 2015 (UTC)
 * I was more looking for one defined in 3 dimensions by a differential equation (i.e. not a discrete case), but this provides enough intuition, thanks.--Jasper Deng (talk) 22:45, 11 December 2015 (UTC)