# One Dimensional Maps - Demo 1

## Iterating the Map

The basic scheme of iterating one-dimensional maps is shown by the applet. Hitting "Step" will
perform one iteration of the map.

From the initial value *x*_{0} of *x* the value of *y=F(x*_{0}) (a vertical "step" on the plot) gives the value of the next iterate. The horizontal "tread" on the plot graphically changes this *y* to an *x* value i.e. gives *x*_{1}. Successive iterations are then given by repeating the "step-tread" sequence.

For *a=2.8* repeatedly hitting step will eventually lead to *x* converging onto a value that no longer changes under iteration. Such a point is known as a *fixed point*. Any fixed point is located at the intersection of *F(x)* with the diagonal line. After stepping the iteration a few times, you can watch how the iterations behave by by hitting *Start* to start a running iteration. (Set the *Speed* to a convenient value.) To study the behavior for different initial conditions click somewhere in the plot while the iterations are running to start a new iteration with initial condition given by the value of *x* at the mouse position.

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Last modified Sunday, November 28, 1999

Michael Cross