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Stream: community: our work

Topic: Isomorphism of iterated functions

Daniel Geisler (Dec 20 2023 at 07:13):

I have finished up my work with fractional iteration, including Mathematica code which is now in the Wolfram Function Repository. Most of what I have discovered in my research can be unpacked from the following identity for $H$. My work has been influenced by From Finite Sets to Feynman Diagrams and an understanding of structure types.

My focus now is learning category theory and writing proofs. I have written a proof for
$\mathcal{P}[n] \cong \frac{d^n}{dx^n}f(g(x))$ and I now need to prove $\mathcal{K}[n] \cong \frac{d^n}{dx^n}f^t(x)$.

A family of structure types

$\mathcal{I}=(1,2,3,\cdots)$

$\mathcal{P}=MSET(\mathcal{I})$ A000041

$\mathcal{S}=SET(SET_{\ge 2}(\mathcal{Z}))$ A000110

$\mathcal{K}=\mathcal{Z} +MSET_{\ge 2}(\mathcal{K})$ A000669

$\mathcal{L}=\mathcal{Z}+SET_{\ge 2}(\mathcal{L})$ A000311

Daniel Geisler (Dec 20 2023 at 07:14):

$B_{n,k}$ is a partial Bell polynomial.

${d^n \over dx^n} f(g(x)) = \sum_{k=1}^n f^{(k)}(g(x))\cdot B_{n,k}\left(g'(x),g''(x),\dots,g^{(n-k+1)}(x)\right).$

Set $g(x)=f^{t-1}(x)$

$H(0,t)=L$, where $L$ is a fixed point

$H(1,t)=f'(L)^t$ the Lyapunov multiplier, denoted $\lambda$, with $\lambda \ne 0$.

$H(n,t)=\sum_{r=0}^\infty(\sum_{k=1}^n \frac{f^{(k)}(L)}{k!} B_{n,k}(H(1,t-1),\ldots, H(n-k+1,t-1)))^r$

$f^t(x)=\sum_{k=0}^\infty\frac{1}{k!} H(k,t) (x - L)^k$

Mathematica code

FractionalIteration function

FractionalIteration[fiFunction_, fiTime_, fiTerms_Integer : 4,
    OptionsPattern[]][fiPosition_] := Module[{f, t, x, fp, r, h},
   fp = OptionValue["FixedPoint"];
   h[0] = fp;
   h[1] = f'[fp]^t;
   If[OptionValue["ForceParabolic"],
    f'[fp] = 1;
    ];
   Do[
    h[max] = First[r[t] /. RSolve[{r[0] == 0, r[t] == Sum[
            Derivative[k][f][fp] BellY[max, k,
              Table[h[j] /. t -> t - 1, {j, max}]], {k, 2, max}]
           + f'[fp]*r[t - 1]}, r[t], t]],
    {max, 2, fiTerms}];
   Sum[1/k! h[k] (x - fp)^k, {k, 0, fiTerms}] /. f -> fiFunction /.
     t -> fiTime /. x -> fiPosition
   ];