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