Bifurcation diagrams can be created for other functions. Consider the
bifurcation diagram of x^{2} + µ (choose the x^{2} + µ choice
on the bifurcation diagram plotter and plot it).
Below is a plot of the
Mandelbrot set and right underneath is the bifurcation diagram for x^{2} + µ.
For the Mandelbrot set, µ and x are both complex valued. The window
we look at is based on the parameter µ = a + i b where
2.05 <= a <= 0.55 and 1.3 <= b <= 1.3.
The Mandelbrot set was obtained by determining which points in this window
escape to infinity (colored) and which do not (colored black). Points
escaping to infinity fastest are colored red, while the slowest are violet.
The bifurcation diagram amounts to plotting x^{2} + µ for
2.05 <= a <= 0.55 and b = 0 (i.e. µ Real), and
iterating starting with x = 0. The bifurcation diagram displays points
that do not escape to infinity (i.e., periodic and fixed points),
so it corresponds to black regions
of the Mandelbrot set.
It is interesting to note that one can determine some of the periods of
the periodic points of the Mandelbrot set by comparing it to the bifurcation diagram
of x^{2} + µ. The peroid 1 points of the bifurcation diagram corresponds
to the interior of the cardiod area, the circle to the left of the cardiod is
period 2, the next "circle" to the left is period 4, and the little black spot
on the spike corresponds to period 3
