Bifurcation Diagram of x2 +

Bifurcation diagrams can be created for other functions. Consider the bifurcation diagram of x2 + (choose the x2 + 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 x2 + .

For the Mandelbrot set, and x are both complex valued. The window we look at is based on the parameter = 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 x2 + for -2.05 <= a <= 0.55 and b = 0 (i.e. 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 x2 + . 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