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Iterated functions

Iterated functions

Introduction

Given a function such as:
We can call this function for x = 1:
This is called the first iterate of x.
We can then call the function again on the result:
Which is the same as:
It is called the second iterate of x.
We can generalize the function to:
Or even more generally:
The functions f¹(x) to fⁿ(x) are called the iterates of x for f.
- It is also called the orbit of x
We also use subscripts for the iterates of x: $\begin{align} x_0 &= \tex$

Behavior of iterated functions

Constant

Some functions always produce the same value:
Produces:
n 0 1 2 3 4
fⁿ 1 1 1 1 1

Oscilating

Some functions cycle between a few values:
Produces:
n 0 1 2 3 4
fⁿ 1 -2 1 -2 1

Converging

Some functions converge to a single value: $f(x) = \frac{x}{2} + \fra$
Produces:
n 0 1 2 3 4
fⁿ 10 5.2 2.98 2.16 2

Random

Some functions produce seamingly random values:
Produces:
n 0 1 2 3 4
fⁿ 0.3 1.11 3.21 2.54 3.71

Expanding

Some functions keep on growing forever:
Produces:
n 0 1 2 3 4
fⁿ 2 4 16 256 65536

Graphing iterated functions

Let's start with the converging function: $f(x) = \frac{x}{2} + \fra$
The function converges when:
If we plot the function, along with the line:
Then the point where they intersect will be where the function converges:
The function converges at x_n = 2

Graphing the iterates

If we start at point (x₀, x₀)
We can draw a line vertically until it meets f(x)
It will meet at the point (x₀, f(x₀)
From this point we can draw a line horizontally until it meets the line: y = x
it will meet at the point (f(x₀), f(x₀))
This is the point (x₁, x₁)
We can continue this process until x_n = x_{n - 1}:

Graphing oscilating functions

Given the function:
If we start at point (1, 1) and draw arrows between the two lines:
Then we end up back where we started, at (1, 1)