Fractional iteration of a function

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SUMMARY

The forum discussion centers on the concept of fractional iteration of the function \( f(x) = 6x^5 - 15x^4 + 10x^3 \) defined on the interval \( x \in [0, 1] \). The user seeks to understand how to smoothly transition from the linear function \( x \) to its iterated forms \( f(x) \) and \( f(f(x)) \) using fractional iteration techniques. The discussion highlights the challenge of expanding higher-derivative terms in the Taylor series, which is crucial for achieving the desired smooth transition. The user has a background in calculus and linear algebra but feels rusty in applying these concepts.

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  • Understanding of fractional iteration concepts
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  • Knowledge of calculus, specifically derivatives and the chain rule
  • Basic linear algebra principles
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  • Study Taylor series expansion techniques for higher derivatives
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Lord Crc
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I have the following function \begin{align*}<br /> f(x) &amp;= 6x^5 - 15x^4 + 10x^3 &amp; x &amp;\in [0, 1]<br /> \end{align*} and I found that by recursively applying it, that is f(f(x)) etc, I can get new functions with the same s-like shape but steeper slope.

I was curious if there was a way to smoothly go from x (ie a line) to f(x) to f(f(x)) and beyond, and some Googling told me this is called fractional iteration.

However the Wikipedia page didn't leave me with enough clues for finding the appropriate series. In particular I'm struggling to see how to expand the higher-derivative terms in the Taylor expansion in step 4. Any help with this would be appreciated.

FWIW I did go to university and had a fair share of calculus and linear algebra, but it's some 7+ years ago and I'm rusty these days. Oh and this is purely for fun, not homework.
 
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