12/17/2022, 07:37 PM
Fractional iterated function.
Note each of the following are combinatorically based - finite inputs result in a finite output. Polynomial, exponential, derivatives, summations, partitions, Bell polynomials, recursive Bell polynomials are all combinatorial. The function to be iterated is homomorphic.
So the following fractional iterated function is convergent.
\[H(0,t)=L\], where \[L\] is a fixed point
\[H(1,t)=f'(L)^t\] the Lyapunov multiplier, denoted \[\lambda\], with \[\lambda \ne 0\].
\[H(n,t)=\sum_{r=0}^\infty(\sum_{k=1}^n \frac{f^{(k)}(L)}{k!} B_{n,k}(H(1,t-1),\ldots, H(n-k+1,t-1)))^r\]
\[f^t(x)=\sum_{n=0}^\infty\frac{1}{k!} H(k,t) (x - L)^k\]
Note each of the following are combinatorically based - finite inputs result in a finite output. Polynomial, exponential, derivatives, summations, partitions, Bell polynomials, recursive Bell polynomials are all combinatorial. The function to be iterated is homomorphic.
So the following fractional iterated function is convergent.
\[H(0,t)=L\], where \[L\] is a fixed point
\[H(1,t)=f'(L)^t\] the Lyapunov multiplier, denoted \[\lambda\], with \[\lambda \ne 0\].
\[H(n,t)=\sum_{r=0}^\infty(\sum_{k=1}^n \frac{f^{(k)}(L)}{k!} B_{n,k}(H(1,t-1),\ldots, H(n-k+1,t-1)))^r\]
\[f^t(x)=\sum_{n=0}^\infty\frac{1}{k!} H(k,t) (x - L)^k\]
Daniel