In this lesson, we will study the behavior of the iterates of a function using the graph of the function. Instead of starting with just one value, and then using a computer or calculator to find the orbit of this starting value, we will use the graph of the function to find the orbits of different starting values.

In the previous lesson, we touched on some of the ways that orbits can go. We saw examples where the orbits:

kept getting closer and closer to a certain value, for example, iterating x = 0.5 with the function

went back and forth between two values, for example iterating x = 1 with the function

started out moving around a little, and then settled on one point, for example, iterating x = 1 with the function

Some people have a very keen intuition for how orbits will work out, and can spot starting values that will give unusual orbits just by looking at the formula for the function that will be used to do the iterations.

Most of the time, though, it is quite hard to tell how orbits will work out just from the formula for the function alone. In these situations, analyzing the graph of the function can reveal how the orbits will go.

Why Orbits Anyway?

Some people find the subject of iterated functions absolutely fascinating, and worthy of study for its own sake. However, this is supposed to be a course on chaos and fractals, so whay all the business with all this iterating and orbits, anyway?

One image that is very popular in pouplar books and articles on the subject of chaos is the Julia set. Julia sets come in many shapes and sizes, one famous shape and size is called "The Rabbit,"

Many people agree that these images possess a mysterious beauty. Some might be surprized to learn that this beauty derives from an operation as seemingly mundane as iteration of mathematical functions.

In lesson 1, the orbits did one of the following:

1. The escaped, i.e. the orbits that went off to infinity.

2. They were attracted to a point, i.e. the orbits that kept getting closer and closer to a value, the orbits that moved around for a bit and then settled down, and the orbits that never went anywhere.

3. Repeated a cycle periodically, i.e. the orbits that went back and forth between two values.

One way to categorize starting points is to determine whether their orbits escape (i.e. option 1 above) or not (options 2 and 3 above). We could make a picture by coloring starting points one color if their orbit escapes, and another color if the orbit doesn't escape.

As an example, consider the function g(x) = x². From lesson 1, the starting points are the real numbers, and the orbit escapes if the starting point is greater than 1, or less than -1. Coloring starting points whose orbit (using the function g(x) = x² to do the iteration) escapes blue, and starting points whose orbit (using the function g(x) = x² to do the iteration) red gives the following picture:

The really important feature of this picture is the red and blue line. Notice that this is a line, because real numbers are used as starting points, and the collection of all real numbers is a one-dimensional line.

This particular coloring of the starting points isn't too exciting, but the idea might have some possibilities. For example, using a more complicated function than just g(x) = x² might lead to more complicated and interesting orbits, and to a more complicated and interesting coloring of the starting points for the iteration. This is the case. Another possibility exists; using a fairly straight-forward function to do the iteration, but a more complicated set of numbers to do the calculation with. After the real numbers, there are the complex numbers, so we could try them.

If you have never studied the complex numbers, feel free to click here to learn a little about them.

If we use complex numbers instead of real numbers, then instead of just a one-dimensionsal line of starting points for the iteration, we will have a two-dimensional plane of starting points to consider. So, if starting points for iteration are colored white if ther orbits escape, and black if their orbits don't escape, then we'll have a two-dimensioanl picture, rather than a one dimensional picture like the red and blue one above.

Using the function:

to do the iterations, and coloring starting points white if their orbit escapes, and black if their orbit doesn't escape gives the following picture:

This picture is immediately recognizable as the silhouette of "The Rabbit." The extra colors of "The Rabbit" are added by coloring starting points different colors depending on how quickly their orbit escapes.

Iteration of Mathematical Functions

Remember from the first lesson how orbits are generated; you have a function, and a number as a starting value, and then you iterate. Take, for example, the (positive) square root function,

and a starting value of x = 100. With the positive square root function, iteration means taking square roots again and again.

This kind of approach is feasible if you want to figure out the orbit of just one or two different starting points. If you want to figure out the orbits of many starting points, then this kind of method is very labor intensive.

An excellent device for studying the behavior of a function at many points is the graph of the function. For example, the graph of

for 0 =< x =< 25 is shown below.

The next iterate would be . Graphing the second iterate for

0 =< x =< 25 gives the graph shown below.

Plotting both the first iterate (i.e. f(x)) in red and the second iterate (i.e. f²(x)) in blue shows how the orbits change from the first iterate to the second iterate for all the starting values from zero to 25 at the same time.

The changes that occur between the red graph (the first iterate) in the blue graph (the second iterate) already hint at what the orbits will do:

This behavior could be confirmed by drawing the graphs of higher iterates of the function.

What were the results of your calculator investigations of the orbits of ? Do they agree with the result suggested above from the graphical analysis.

Try using graphical analysis to determine what the orbits are when the function used to do the iterating is

So, the graphs of the iterates of the function can be used to study what the behavior of the orbits of many different starting points all at the same time. This process is still pretty labor intensive (the graphs of all the iterates of the function need to be plotted for example), but it is better than doing the loads of calculations that were required in the last lesson.

A third possibility exists; using graphs to study the behavior of the orbit of just one starting point at a time.

Using Graphs to Study a Single Orbit.

Right now, this method will simply be described. You are invited (there is a thinking opportunity on this later) to consider what the procedure described below has to do with iterations of functions. The procedure is illustrated by showing how it could be used to study the orbit of the starting point x = 0.2 when the function is used to do the iterations.

Step 1 Sketch a graph of the function (red in the diagram below) and a graph of y = x (blue on the diagram below) using the same set of axes.

Step 2 Since a starting value of x = 0.2 is used, locate the point (0.2, 0.2) on the blue line. Draw a vertical line (shown in yellow), heading towards the red curve, from the point (0.2, 0.2) until you hit the red curve.

Step 3 From the place where the yellow line meets the red curve, draw a horizontal line (shown as a green dotted line) heading towards the blue line, until you hit the blue line. The coordinates of the point that you hit on the blue line are equal to the first iterate of the starting point (x = 0.2) when the function is used to do the iterations. In this case, the point you reach has coordinates (0.64, 0.64). It is easy to check that g(0.2) = 0.64 with a calculator, or by hand.

Step 4 From the place where the green dotted line hit the blue line, start drawing a vertical line (shown in gray), heading for the red curve, until you hit the red curve.

Step 5 From the place where the gray line hit the red, start drawing a horizontal line (shown in pink), heading for the blue line, until you hit the blue line. The coordinates of the point where you hit the blue line are equal to the second iterate when you start at x = 0.2, and use the function to do the iterations. In this case, the point that you hit on the blue line is (0.9216, 0.9216), so g²(0.2) = 0.9216.

Step 6 Continue in this fashion to find higher iterates.

Can you use this form of graphical analysis to show that when the function is used to do the iterations,

Proposition Suppose that you are doing graphical analysis, and that you are using a function k(x) to do the iteration with. Suppose that there is an x-value (say x₀) where the graph of y = k(x) and the graph of y = x intersect. Then, if x₀ is used as a starting point, the orbit never goes anywhere.

Using Graphical Analysis to Identify Patterns in Iteration

There are several features of orbits that graphical analysis can be used to illustrate very beautifully and dramatically.

Attracting and repelling fixed points

You might have noticed an interesting feature of the points whose orbits don't go anywhere (because they always stay fixed in one place, they are called fixed points). The points nearby some fixed points move away from the fixed point as you iterate, whereas for other fixed points, the points nearby the fixed point move towards the fixed point as you iterate.

For example, when you use the function g(x) = x² to do the iterations, the fixed points are x = 0, and x = 1. The starting points near zero (in fact, all the starting points between -1 and 1) move towards zero when you iterate using the function g(x) = x². However, the points around x = 0 (in fact, all possible starting points except x = 1) move away from x = 1 when you iterate. The starting points between x = -1 and x = 1 move towards zero, whereas the starting points x < -1 and x > 1 move towards infinity.

You can see this behavior by using graphical analysis on different starting points. For example, if take a starting point of x = 0.75, then first few steps of the graphical analysis resemble the following graph.

As the path of little arrows shows, when you iterate the starting point x = 0.75, the orbit moves towards x = 0, and away from x = 1. A fixed point like x = 0 that seems to attract orbits to it is called an attracting fixed point.

Using graphical analysis with a starting point larger that x = 1 (say x = 1.5) give a picture resembling the following graph.

The path of little arrows starting at the point (1.5, 1.5) shows how the orbit moves away from x = 1 as the iterations are done. A fixed point like this, that orbits move away from, is called a repelling fixed point.

Periodic cycles

You might remember from earlier in the lesson that the Julia set was constructed by coloring the complex plane. A point was colored white if, when it was used as a starting point, its orbit went off to infinity, and was colored black if, when it was used as a starting point, the orbit didn't go off to infinity.

When the complex function is used to do the iterations, the picture that you get is,

In the previous section on attracting fixed points, one of the ways that the orbit can avoid going off to infinity was introduced. Instead of going off to infinity, the orbit is attracted to an attracting fixed point.

In all of the examples that we have considered, orbits that were attracted to an attracting fixed point always moved towards the fixed point. For example, when a starting value between 0 and 1 was selected, and the function is used to do the iterations, then the orbit always moves closer and closer to 1 with each successive iteration. Does this always have to be the case? Is it possible to have an orbit that starts moving towards a fixed point, and then later on turns around and starts moving away from the fixed point? Give examples or arguments to support your conclusions.

One of the interesting points from the last thinking opportunity is the possibility of orbits that do not just go off to infinity, and yet are not attracted to attracting fixed points. When the function

is used to do the iterations, the starting point of x = 1 (or x = -1) show this behavior. Graphical analysis can be used to illustrate the phenomenon.

In the graphical analysis, the path of the arrows forms a closed circuit or cycle. This phenomenon is called a periodic cycle. The points x = 1 and x = -1 are called periodic points.

Attracting and Repelling Periodic Points

Fixed points were called attracting if nearby starting points had orbits that were attracted to the fixed point. Fixed points were called repelling if nearby starting points had orbits that moved away from the fixed points.

Periodic points can be classified as attracting and repelling in exactly the same sort of way. Just as graphical analysis could be used to investigate whether a given fixed point was attracting or repelling, graphical analysis can be used to investigate whether periodic points are attracting or repelling.

Are x = 1 and x = -1 the only periodic points that occur when the function

is used to do the iterations?

Stable and Unstable Orbits

An orbit is described as stable if, when the starting point is changed slightly, the resulting orbit behaves much the same as it did before.

For example, when the function

is used to do the iterations, all orbits that start at non-zero starting points are stable. When f(x) is used to do the iterations, all orbits that start with non-zero starting values go towards the attracting fixed point '1.' So, if you start with a starting value of x = 2, the orbits with starting values pretty close to 2 will all look very much like the orbits that has starting value x = 2.

However, when the function

is used to do the iterations, the orbit with starting value x = 1 is not stable. The orbit with starting value x = 1 doesn't go any where, it always just stays with value '1.' If you decrease the starting value slightly (say to x = 0.99999999999999999999999999999) then the orbit doesn't just stay fixed in one spot, instead, it gets closer and closer to zero. Alternatively, if you increase the starting value slightly (say to x = 1.00000000000000001) then the orbit doesn't stay fixed in one spot, either. Instead, the orbit goes off to infinity.

Is there any relationship between the stability of an orbit of a periodic point, and whether the periodic point is a repelling periodic point or an attracting periodic point?