General form for this iterated function?

Asked Feb 05 '23 at 18:08

Active Feb 06 '23 at 03:45

Viewed 55 times

The function is $f(x)=3x^2-2x^3$

I want to iterate this function like this $f(f(f(x)))$.

However I can't tell if there is a general form for $f^n(x)$.

The coefficients explode by the 2nd iteration and I can't discern any kind of pattern.

I'm way out of my depth here but if anyone could help that would be great. Also if anyone knows a better way to blend between this function and approximately a step function that would be very helpful.

$f(f(x)) = 16x^9-72x^8+108x^7-42x^6-36x^5+27x^4$

$f(f(f(x)) = -8192 x^{27} + 110592x^{26} - 663552 x^{25} + 2304000 x^{24} - 5004288 x^{23} + 6676992 x^{22} - 4368384 x^{21} - 1347840 x^{20} + 5295456 x^{19} - 3933456 x^{18} + 94176 x^{17} + 1641384 x^{16} - 919656 x^{15} + 23436 x^{14} + 148392 x^{13} - 69066 x^{12} + 26568 x^{11} - 2916 x^{10} - 5832 x^{9} + 2187 x^{8}$

Thanks for the help everyone. I ended up finding the generalized higher order smooth step polynomials on this Wikipedia page

Close enough to what I was looking for.

edited Feb 06 '23 at 03:45

asked Feb 05 '23 at 18:08

Bailesdtd

Can you give us the results you have computed so far? – StiftungWarentest Feb 05 '23 at 18:18
@StiftungWarentest Yes. I have edited the post. – Bailesdtd Feb 05 '23 at 19:04
1

Write x^{27} to make both digits be raised: $x^{27}.$ As you have not, without the braces, you get $x^27.$ – md2perpe Feb 05 '23 at 19:33
Try \begin{align} f(x)&=x^2(3-2x),\ f^{\circ 2}(x)&=x^4(3-2x)^2(3-2x^2(3-2x)),. \end{align} It is better to write $f^{\circ n}$ to avoid confusion. It is also not quite clear to me why there should be a nice pattern. – Kurt G. Feb 05 '23 at 20:26
Thanks for the formatting tips. I'll remember for my next post. – Bailesdtd Feb 06 '23 at 03:49

General form for this iterated function?

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