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One-Dimensional Iteration

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, c. Below the picture is an explanation.

Quadratic Map

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 x to iterate, the program draws a vertical line to first iterate, (x,f(x)). Then it draws a horizantal line to the point (f(x),f(x)) 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(x), f(f(x))), the second iterate. Repeating this process gives all the iterates of x.

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.


One-Dimensional Iteration applet written by Evelyn Sander with the help of Packer Layout written by Daeron Meyer.

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Created: Mar 26 1996 --- Last modified: Fri Apr 5 10:53:29 1996