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# One-Dimensional Dynamical Systems

## Part 3: Iteration

The function

*f*(*x*) =
*x*
(1 - *x*)
is known as the *Logistic map*. The Logistic map depends on the
parameter . As we
already saw in the
*Introduction*, the
dynamics of the map depends on this parameter. However, the difference
in the dynamics is not qualitatively significant for every different
. In dynamical
systems theory, we like to think of the whole set of maps depending on
as one
object. To express this idea, we speak of the one-parameter Logistic
*family*. We discuss the dynamics of such a family by
showing the qualitatively different types of possible dynamics, and by
showing in which intervals of the parameter space these dynamics
occur. The one-parameter Logistic family is our object of study.

### Visualizing iterations

The dependency on
and *x*
is nicely illustrated by the Java Applet
Iteration of the Logistic Map of Andy Burbanks. Use this applet to
study how the graph of the map changes as you increase
, and what
happens with the orbit of the chosen initial point. For fixed
do the orbits
of all the points in the interval [0,1] behave the same?

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Created: Apr 1 1998 ---
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