The Quadratic Map
Here is a one-dimensional quadratic parametrized curve x^2+c. When you
click on any point in the graph window, you see the graphical iterates
of that x value. The righthand scrollbar changes the parameter. Below
the picture is an explanation.
About Graphical Iteration
Given a point x, f(x) is its iterate under function f. The dynamics of
f is understanding what happens to points after successive iterates.
Graphical iteration is a visual technique to see where points go under
iteraton. Above you see a graph of your function y=f(x) and a graph of the
identity function y=i(x). When you choose a value z to iterate, the
program draws a vertical line to first iterate, (z,f(z)). Then it
draws a horizantal line to the point (f(z),f(z)) on the graph of the
identity function. This allows a second iteration, because drawing a
vertical line from here to the graph of f gives the point (f(z),
f(f(z))), the second iterate. Repeating this process gives all the
iterates of z.
What to look for
Notice that there is an attracting fixed point at c=0. As c decreases,
the fixed point ceases to be attracting and a period two orbit
forms. This is called a period doubling bifurcation. Move the scroll
bar to see this bifurcation occur. As c decreases further, the period
two orbit ceases to attract and there is an attracting period four
orbit. This process continues, with each period 2^n orbit ceasing to
attract exactly when a period 2^(n+1) orbit forms. This is called a
period doubling cascade. Finally, there are points of all
periods. Around c=-1.92, there is chaos.
This was written with the help of Packer
Layout by Daeron Meyer.
Last modified: Thu Feb 15 16:24:15 1996