One-Dimensional Iteration

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.

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.

Acknowledgements

This was written with the help of Packer Layout by Daeron Meyer.
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Evelyn Sander<sander@geom.umn.edu>