# Basic Concepts of Chaos

## Iteration

Iteration means repeating a process over and over. Countless examples of iteration can be found in the real-world:

• To walk, you simply iterate the process of taking a step.
• To eat french fries, you iterate the process of putting a fry in your mouth.
• To pump up a flat bicycle tire, you iterate the process of pushing the pump down and up.
In mathematics, iteration means repeating a function over and over. Usually a seed, or initial value, must be specified. The seed is generally denoted as x0. Each subsequent x-value is denoted with incrementing subscripts, x1, x2, x3, x4, . . . so x4 is the x-value after 4 iterations (as xn is the x-value after n iterations). One way to denote "the nth iteration of x0" is fn(x0). So f2(x0) = f(f(x0)), f3(x0) = f(f(f(x0))), and so on. The basic concept of iteration (as well as the notation) is demonstrated in the following chart, which shows the iteration of the function f(x) = x2 using 2 as x0.

 n xn-1 different ways to show fn(x0) xn 1 2 f(x0) 4 2 4 f(x1) or f2(x0) or f(f(x0)) 16 3 16 f(x2) or f3(x0) or f(f(f(x0))) 256 4 256 f(x3) or f4(x0) or f(f(f(f(x0)))) 65536

## Orbits

The orbit of a seed is defined as the sequence of values (points on the number line) that satisfy the equation xn = fn(x0). Put simply, the orbit of a seed x0 is {x0, x1, x2, x3, x4, . . . }. In the previous example of f(x) = x2 with x0 = 2, the orbit is {2, 4, 16, 256, 65536, . . . }.

### Different types of orbits

Fixed - This type of orbit is exhibited by x0 values that satisfy the equation x0 = f(x0). If a point is fixed, its orbit remains constant, such as with f(x) = x2, 1 is a fixed point whose orbit is {1, 1, 1, 1, . . . }.

Eventually Fixed - Points that show this orbit behavior do not start off as fixed, but after n iterations, there orbits become fixed. For example, with f(x) = x2, -1 is an eventually fixed point, because its orbit is {-1, 1, 1, 1, . . . ).

Periodic - This type of orbit is shown by x0 values that satisfy the equation x0 = fn(x0) where n is any positive integer. The lowest possible n for such an orbit is called the prime period of the orbit. If an orbit is prime period n, it can also be referred to as a period-n orbit. An example of a period-2 orbit can be seen with f(x) = x2 - 1, using -1 as x0; the orbit is {-1, 0, -1, 0, -1, 0, . . . }, alternating between -1 and 0.

Eventually Periodic - As with eventually fixed orbits, eventually periodic points do not start off as periodic; however, after n iterations, there orbits become periodic. An example of an eventually periodic point can be seen in f(x) = 1 - x2, using -1 as x0; the orbit is {-1, 0, 1, 0, 1, 0, 1, . . . }.

Increasing - Points behaving in this type, as the name implies, constantly increase when iterated through the function. An increasing orbit can be seen on the function f(x) = x2 for any seed greater than 1. Using 5 as the seed, for example, the orbit is {5, 25, 625, 390625, 152587890625, . . . }.

Decreasing - This type of orbit is the counterpart to the increasing orbit. Decreasing points always lower in value, when iterated through the function. This type of orbit is exhibited on the function f(x) = 3x using any seed lower than 0. for instance, using -2 as the seed produces the orbit {-2, -6, -18, -54, -162, . . . }.

Chaotic - These are points that do not fit into any of the above categories and whose orbits seem to 'jump around' randomly on the number line. For an example of a chaotic orbit, see the following chart, which shows some of the orbit of x0 = 4 on f(x) = 4x(1 - x):

 n xn 0 .6 1 .96 2 .1536001 3 .5200284 4 .9983954 5 6.40793 x 10-3 ... ... 95 .1560364 96 .5267562 97 .9971364 98 1.142154 x 10-2 99 4.516437 x 10-2 100 .1724982
As you can see, the orbit bounces around in a complex (and seemingly random) pattern.

## Attraction and Repulsion

As we learned above, fixed points have orbits that do not change. These points also demonstrate some interesting properties, among which are attraction and repulsion. Depending on the slope of a fixed point, it can be attracting, repelling, or neutral. These three different characteristics are named for the effect of the fixed point on other nearby points:

• Attracting fixed points, found where the absolute value of the function's slope is less than 1, pull nearby points closer. That is, after a few iterations, a nearby point (used as the seed) will be pulled very close to the attracted fixed point.
• Repelling fixed points, found where the absolute value of the function's slope is greater than 1, push nearby points away. After a few iterations, a nearby point (used as the seed) will be pushed far away from the repelling fixed point.
• Neutral fixed points are found where the function's slope is equal to 1 or -1. Despite the misleading name, these points can attract, repel, or do both. Whether a neutral fixed point will attract, repel, do both, or do neither depends on the specific case. Some different cases of neutral fixed points are illustrated, along with cases of attracting and repelling points, in the following chart.

 Type of fixed point Attracting Repelling Neutral: Both Function f(x) f(x) = 2x(1 - x) f(x) = 2x(1 - x) f(x) = x - x2 Fixed point .5 0 0 Nearby point (x0) .1 .7 -.3 .4 -.2 .3 x1 .18 .42 -.78 .48 -.24 .21 x2 .2952 .4872 -2.7768 .4992 -.2976 .1659 x3 .4161139 .4996723 -20.97484 .4999987 -.3861658 .1383772 x4 .4859263 .4999998 -921.8373 .5 -.5352898 .1192289 x5 .4996039 .5 -1701412 .5 -.8218249 .1050134 x6 .4999997 .5 -5.79x1012 .5 -1.497221 .09398559 x7 .5 .5 diverges to negative infinity .5 diverges to negative infinity converges to 0

## Graphical Analysis

Graphical Analysis is a method of graphing that helps us understand the dynamics (different orbits) of a one-dimensional system. With graphical analysis, we use the Cartesian graph of a function f(x) and the diagonal line y = x to determine where a seed, when iterated through the function, will go on the real number line.

### How to do it:

First, graph any one-dimensional function f(x). Then, graph the diagonal line y = x on the same plane. The points where the two graphs intersect are the fixed points, where f(x) = x. To determine where the orbit of any seed goes, find the seed on the graph of f(x) at the coordinates (x0, f(x0)). Draw a horizontal line from f(x) to y = x. Then, draw a vertical line from y = x to f(x); this is the next point in the orbit of x0. Repeat these last two steps a few times, and you will (usually) be able to understand the nature of the orbit (that is, whether it is fixed, periodic, increasing, decreasing, etc.).

## Bifurcations

To learn about bifurcations, one must study a family of functions. A family is a group of functions with similar graphs, but whose formulas differ in one parameter value. For instance, the quadratic family consists of every function f(x) = x2 + p where p is a parameter that varies in different functions. You can see that decreasing the value of p causes the graph of f(x) to be lower, and increasing p causes the graph of f(x) to be higher. As the parameter changes, the graph 'moves' (in some cases it can be stretched, shrunk, and even flipped); this 'motion' can lead to different dynamics within a family of functions. So if for some function f(x) that takes one parameter p, f(x) with p = 2 might have one attracting fixed point, and f(x) with p = 1.5 might have two repelling fixed points.

Now that you know that parameters can affect the dynamics of a function, bifurcations will be easier to understand. A bifurcation is basically the splitting apart of a fixed point into two new fixed points when the parameter crosses a certain value. Although there are many different types of bifurcations, we will only deal with one, the saddle node bifurcation. A function undergoing a saddle node bifurcation when the parameter p is equal to q, suits the following conditions:

1. The function f(x) has no fixed points when p > q.
2. The function f(x) has one fixed point when p = q.
3. The function f(x) has two fixed points when p < q.
or
1. The function f(x) has no fixed points when p < q.
2. The function f(x) has one fixed point when p = q.
3. The function f(x) has two fixed points when p > q.
As you can see, these two possibilities simply represent opposite cases of the saddle node bifurcation (whether p is increasing or decreasing). For a much needed graphical description of the concept of the saddle node bifurcation,