Up: JAVA Gallery of Interactive On-Line Geometry
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.
Acknowledgements
One-Dimensional Iteration applet written by
Evelyn Sander with the help of
Packer Layout
written by Daeron Meyer.
Up: JAVA Gallery of Interactive On-Line Geometry
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Created: Mar 26 1996 ---
Last modified: Fri Apr 5 10:53:29 1996