The Chaos
Mathematics in the age of the computer

1. Mathematical Experiments

search icon

1.2 Bifurcation

All diagrams rendered with 1‑D Chaos Explorer.

A more intuitive approach to orbits can be done through graphical representation using the following rules:

  1. Draw both curves on the same axes. Pick a point on the x-axis. This point is our seed.
  2. Draw a vertical straight line from the point until you intercept the parabola.
  3. Draw a horizontal straight line from the intercept until you reach the diagonal line.
  4. 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).

webThis graph shows the simple fixed-point attractive behavior of the parameter value c = ¼ 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 ½. Further examination shows this approach to be asymptotic.
webIn this graph, the parameter value was set at c = −¾. 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.
webWhen c = −13/16 the orbit settles into a two-cycle, alternating between −¾ and −¼.
webHere we see a four-cycle. When c = −1.3, the orbit oscillates over the values −1.2996224637, 0.3890185483, −1.1486645691, and 0.0194302923. This one settles down rather quickly. After only 100 iterations, it already looked complete.
webThis 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.
webIf 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

ƒ: 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).

bifurcationHere we see the full bifurcation diagram. Parameter values outside of the range [−2, ¼] were not included as all of their orbits go to off infinity. Note how the single attracting fixed point bifurcates repeatedly and then becomes chaotic. Note also the window at c = −1.8. Let's examine these areas in more detail.
bifurcationHere we see a magnification of the period-doubling region. Note successive bifurcations.
bifurcationZooming in on the region in the upper left-hand corner we see a repeat of the large scale structure. The period-doubling region exhibits self similarity, that is, small regions look similar to large regions. This property can be seen in other parts of the diagram.
bifurcationHere we see a magnification of the chaotic regime. Note the windows of periodicity amidst the chaos. Let's zoom in on the largest.
bifurcationThe structure of the window repeats the structure of the overall bifurcation diagram. The period doubling regime is the same but multiplied by three; that is, 3, 6, 12, 24, 48… instead of 1, 2, 4, 8, 16…. Note the window inside each lobe. The perspective is a bit whack as the window covers a region that is taller than it is wide.
bifurcationThis is a magnification centered on the center lobe of the largest window in the center lobe. Note the scale. We have zoomed in 1000 times. This diagram looks astonishingly similar to the original. The more things change the more they stay the same.