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Let $f(x)=a+bx^2$. Define $f_n(x)$ to be the $n$-fold composition of $f$. That is $f_1(x)=f(x)$ $f_2(x)=f \circ f(x)$ $f_n(x)=f \circ f_{n-1}(x), n \ge 2$

Is there a way to find a formula for $f_n$?

I tried to write down $f_2$, $f_3,\ldots$, but I don't see any pattern.

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    [I don't think there's a simple expression for the $n$-th iterate...](http://mathworld.wolfram.com/QuadraticMap.html)2019-04-26

2 Answers 2

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After a simple change of variables, you will be iterating $z^2+c$ for some $c$. There is a simple formula for the $n$th iterate when $c=0$ or $c=-2$. But not otherwise.

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Double-angle formula: $\cos 2\theta = 2\cos^2-1$, so if we write $z=2\cos\theta$, then we get $f_1 = z^2-2 = 2\cos 2\theta$ ; $f_2 = (z^2-2)^2-2 = 2\cos 4\theta$ ; and so on ... $f_n = 2\cos(2^n\theta)$. If you like, put $\theta = \arccos(z/2)$ into these.

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    I sure as peas ain't. ;) I'm noting the cosine/hyperbolic cosine bit can be unified if you switch to Chebyshev's formulation, that's all...2011-09-29
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I don't believe there is a "nice" formula for $f_n$ or even a pattern. Here's my reasoning:

If the graph of $y = a + bx^2$ intersects the line $y=x$, then there can be chaotic behavior in the values $f_n(x)$ for general $x$. See the neat animation on the Wiki article for "cobweb plot":

http://en.wikipedia.org/wiki/Cobweb_plot

Hope this helps!

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    Of course, there are some special cases of note. If $a = 0$, then it's fairly easy to find the general formula: $f_n(x) = b^{2^n - 1}x^{2^n}$.2019-02-12