DISCOVERY PROJECT

Iteration and Chaos

The iterates of a function $f$ at a point $x_0$ are $f(x_0)$, $f(f(x_0))$, $f(f(f(x_0)))$, and so on. We write

$x_1 = f(x_0)$   $\color{#08F}{The \; first \; iterate}$
$x_2 = f(f(x_0))$   $\color{#08F}{The \; second \; iterate}$
$x_3 = f(f(f(x_0)))$   $\color{#08F}{The \; third \; iterate}$

For example, if $f(x) = x^2$, then the iterates of $f$ at $2$ are $x_1 = 4$, $x_2 = 16$, $x_3 = 256$, and so on. (Check this.) Iterates can be described graphically as in Figure 1. Start with $x_0$ on the $x$-axis, move vertically to the graph of $f$, then horizontally to the line $y = x$, then vertically to the graph of $f$, and so on. The $x$-coordinates of the points on the graph of $f$ are the iterates of $f$ at $x_0$.


Iterates are important in studying the logistic function

$$f(x) = kx(1 - x)$$

$n$ $x_n$
$\;$0 $0.1$
$\;$1 $0.234$
$\;$2 $0.46603$
$\;$3 $0.64700$
$\;$4 $0.59382$
$\;$5 $0.62712$
$\;$6 $0.60799$
$\;$7 $0.61968$
$\;$8 $0.61276$
$\;$9 $0.61694$
10 $0.61444$
11 $0.61595$
12 $0.61505$

which models the population of a species with limited potential for growth (such as rabbits on an island or fish in a pond). In this model the maximum population that the environment can support is $1$ (that is, $100$%). If we start with a fraction of that population, say $0.1$ ($10$%), then the iterates of $f$ at $0.1$ give the population after each time interval (days, months, or years, depending on the species). The constant $k$ depends on the rate of growth of the species being modeled; it is called the growth constant. For example, for $k = 2.6$ and $x_0 = 0.1$ the iterates shown in the table to the left give the population of the species for the first $12$ time intervals. The population seems to be stabilizing around $0.615$ (that is, $61.5$% of maximum).

In the three graphs in Figure 2, we plot the iterates of $f$ at $0.1$ for different values of the growth constant $k$. For $k = 2.6$ the population appears to stabilize at a value $0.615$ of maximum, for $k = 3.1$ the population appears to oscillate between two values, and for $k = 3.8$ no obvious pattern emerges. This latter situation is described mathematically by the word chaos.


The following $TI-83$ program draws the first graph in Figure 2. The other graphs are obtained by choosing the appropriate value for $K$ in the program.
PROGRAM : ITERATE
: ClrDraw
: $2.6 \to K$
: $0.1 \to X$
: For$(N, 1, 20)$
: $K \ast X \ast (1 - X) \to Z$
: Pt-On$(N, Z, 2)$
: $Z \to X$
: End

  1. Use the graphical procedure illustrated in Figure 1 to find the first five iterates of $f(x) = 2x(1 - x)$ at $x = 0.1$.
  2. Find the iterates of $f(x) = x^2$ at $x = 1$.
  3. Find the iterates of $f(x) = 1/x$ at $x = 2$.
  4. Find the first six iterates of $f(x) = 1/(1-x)$ at $x = 2$. What is the $1000$th iterate of $f$ at $2$?
  5. Find the first $10$ iterates of the logistic function at $x = 0.1$ for the given value of $k$. Does the population appear to stabilize, oscillate, or is it chaotic?
    (a)   $k = 2.1$   (b)   $k = 3.2$   (c)   $k = 3.9$
  6. It's easy to find iterates using a graphing calculator. The following steps show how to find the iterates of $f(x) = kx(1 - x)$ at $0.1$ for $k = 3$ on a $TI-83$ calculator. (The procedure can be adapted for any graphing calculator.)
    $Y1 = K \ast X \ast(1 - X)$
    $3 \to K$
    $0.1 \to X$
    $Y1 \to X$    		 Enter $f$ as $Y1$ on the graph list
    Store $3$ in the variable $K$
    Store $0.1$ in the variable $X$
    Evaluate $f$ at $X$ and store result back in $X$
    Press $\fbox{ENTER}$ and obtain first iterate
    Keep pressing $\fbox{ENTER}$ to re-execute the
    $\;$ command and obtain successive iterates

    $0.27$
    $0.5913$
    $0.72499293$
    $0.59813454435$

    		  

You can also use the program in the margin to graph the iterates and study them visually.

Use a graphing calculator to experiment with how the value of $k$ affects the iterates of $f(x) = kx(1 - x)$ at $0.1$. Find several different values of $k$ that make the iterates stabilize at one value, oscillate between two values, and exhibit chaos. (Use values of $k$ between $1$ and $4$.) Can you find a value of $k$ that makes the iterates oscillate between four values?