Attractor is a combination of two fixed points (ends), the area in between the two rised points which an object moves in cycles, and everything else that happens.

Nonlinearity is a type of mathematical system that generally cannot be solved.

FFIXED POINTS: ATTRACTORS AND REPELLERS The picture on the left is the same one as used on the page on geometric iteration. But this time it will be explained further. The grey line is the reflective iteration mechanism, and the other bolder line is the function in question. Note that this is a linear function, and not a function that will lead to the Feigenbaum fractal. Anyway, the purpose is only to understand the basics of iteration.

A fixed point is the intersection of a function and the diagonal y=x. When the function is non-linear, there can be many fixed points, something that will be taken care of later. On the figure at left, there is one fixed point. The special thing about this fixed point is that successive iteration leads us to this fixed point. The fixed point is therefore called an attractor.

One way to represent iterations graphically is to use the geometric method as the picture on upper left. Another way, which do not require understanding of geometric iteration, is by using so-called time-series. This is simply a graph of the function value after n iterations. N goes along the x-axis, and f.n(x) along the y-axis. An example when f(x)=cos(x) is shown on picture on the left. This graph is typically for all fixed points, and you will not get any dramatically different time-graphs by using another function. Also note that this function is non-linear, but that is not important in this situation. But we can conclude that this is a fixed point.

We shall look at three other examples:

This is another example. But in this case successive iteration does not lead us to this fixed point. Instead we get farther and farther away from the fixed point. We therefore call this fixed point a repeller.

The difference between this graph and the one above, is the slope of the function. In the first example it was between -1 and 0, in this case it is less than -1.

We also note that we in in both cases got a spiral in or out. This is typical to iteration of function with a negative slope m (derivative).

In this third example we have a positive slope and do not have any spiral shape. The fixed point is an attractor, as successive iteration will lead us here from any starting point (try for yourself). The slope is between 0 and 1, just as between -1 and 0 was attractive in the first example too.

The shape looks like a stairs, and this iteration is therefore called staircase in ("in" means attractive).

Here we have a positive slope, and apparently a repelling fixed point. The slope is above 1, just as the fixed point was repelling when it was beneath -1.

The shape is called staircase out.

To sum things up:

m < -1
Repelling - spiral out
-1 < m < 0
Attractive - spiral in
0 < m < 1
Attractive - staircase in
1 < m
Repelling - staircase out

More generally:

|m| < 1
Attractor
1 < |m|
Repeller

We also have another possibility:

m = 0: Either all points are attractive or none are. Not very interesting at all.

The same result can be obtained not using geometric iteration at all. I previously said the behaviour of fixed points could not be understood without geometric iteration. This is not the whole truth. It could be understood in the following way, but it is so much more difficult that it is not recommended as a starting point.

We'll start off easy.

When iterating the function f(x), the fixed points can be found solving this equation:

x = f(x)

This is what we have done when the fixed point was the intersection point of the function and the diagonal y=x, as stated above.

The last part of this page is a proof that gives us the criterias listed above for when we have an attractor or a repeller. It is only an example for an attractor, the same thing can be done for a repeller. It is not necessary to read and understand this proof to step further into the feigenbaum fractal.