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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.
## Acknowledgements

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