I think there is a simple answer to this question:
1.2 Bifurcation
A more intuitive approach to orbits can be done through graphical representation using the following rules:
- Draw both curves on the same axes. Pick a point on the x-axis. This point is our seed.
- Draw a vertical straight line from the point until you intercept the parabola.
- Draw a horizontal straight line from the intercept until you reach the diagonal line.
- Repeat step 2 with this new point.
The following is a series of graphs detailing some of the behaviors described earlier. Because of their appearance, these diagrams are commonly known as
web diagrams (or
cobweb diagrams).
This graph shows the simple fixed-point attractive behavior of the parameter value c = 1/4 for the seed value of 0. Zero will be used as a standard seed for all further diagrams because it is "well-behaved". Note how the orbit moves towards 1/2. Further examination shows this approach to be asymptotic.
In this graph, the parameter value was set at c = -3/4. Note how the orbit approaches the fixed-point attractor from opposite sides. After more than 1000 iterations there is still a visible hole in the center. The orbit hasn't yet reached its final value.This orbit was drawn using a parameter value of c = -1.4015. Although it looks similar to the previous diagram, the iterates never seem to repeat. Instead, they slosh around within bands. Tiny adjustments in initial conditions give orbits that are obviously different. At c = -1.4, the orbit had a period of 32, now the orbit has a period of infinity.
If this isn't chaos, I don't know what is. At c = -1.8, the orbit covers every region of some subinterval of [-2, 2]. This picture shows just a small subset of all the points the orbit will eventually visit.
A way to see the general behavior of the mapping
f: x --> x2 + c
is to plot the orbits as a function of the parameter "c". We will not plot all the points of an orbit, just the most indicative ones. The first several hundred iterations will be discarded allowing the orbit to settle down into its characteristic behavior. Such a diagram is called a
bifurcation diagram as it shows the bifurcations of the orbits (among other things).
Ok...that was stupid.
