Function Repository Resource:

# FractionalIteration

Get the flow of an iterated function at a fixed point

Contributed by: Daniel Geisler
 ResourceFunction["FractionalIteration"][f,t,order][x] returns the power series of the flow associated with a fixed-point.

## Details and Options

For the for the continuous or fractional iteration of a function f(x), dependent on continuous parameter s∈ℝ, fs(x) is called a flow if it satisfies the condition . Here, we work with finite iterations of a function so ResourceFunction["FractionalIteration"] can be considered as a discrete analog to a flow; where s is the integer number of iterations.
The following definitions are helpful for understanding ResourceFunction["FractionalIteration"]:
Super-attracting fixed point - dynamics described by Böttcher's equation. Typically does not produce a flow. f'(p)=0.
Hyperbolic fixed point - flow described by Schröder's equation for ft(x) where |f'(p)n|!=1 and fixed-point p is not a super-attractor.
Parabolic fixed point - flow described by Abel's equation for ft(x) where f'(p)=1.
Rationally neutral fixed point - rationally neutral flow when f'(p) is a root of unity, f'(p)n=1, n∈ℕ.
Irrationally neutral fixed point - irrationally neutral flow when |f'(p)|=1.
ResourceFunction["FractionalIteration"] computes flows of iterated functions including hyperbolic, parabolic, rationally neutral and irrationally neutral.
 Method Automatic method to use FixedPoint p fixed point ForceParabolic False sets first derivitive to 1
Being able to take the flow of maps allows not only tetration to be extended to the complex numbers, but all the higher hyperoperators. See the Neat Examples.

## Examples

### Basic Examples (4)

The general hyperbolic flow power-series with a degree of 2:

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The general hyperbolic flow power-series with a degree of 3:

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The general parabolic flow solution with default degree of 4:

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The flow of the Sin function is an example of parabolic iteration:

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### Properties and Relations (1)

Show that the flow law is satisfied to order 4:

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### Applications (2)

#### Hyperbolic Tetration (2)

Show a hyperbolic tetration of a function:

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Error between flow and iteration:

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Daniel Geisler

## Requirements

Wolfram Language 13.0 (December 2021) or above

## Version History

• 1.0.0 – 07 July 2023