**Up:** *Linear and Nonlinear Behavior*

# One-Dimensional Dynamical Systems

## Part 4: Linear and Nonlinear Behavior

#### Eventual behavior

Let us investigate the Logistic family for
= 3.2. Which
of the fixed points are attracting? Which are repelling? What about
the periodic points?

*Qualitative diagram for
= 3.2.*

The above picture represents the eventual behavior of all points on
the line for =
3.2. With arrows we indicate that the two periodic points are
mapped onto each other. For the other points, we use arrows to
indicate the direction that points go after iteration. Note that a
point *x*, close to one periodic point, will first be mapped
close to the other periodic point before it gets mapped back to the
line segment that contains *x*. This is not indicated in the
picture, because this behavior is already reflected in the behavior of
the periodic points themselves.

- Draw a qualitative diagram for the eventual behavior of all
points for =
2. Your diagram should resemble the picture above, but it does not
need to have exact values for fixed and periodic points. The idea
is to compare pictures qualitatively.

- Draw a similar diagram for
= 3.1. As you
investigate, magnify the region near the fixed point other than
0. Compared to the behavior for
= 2, what has
happened?

- Draw a similar diagram for
= 3.5. What is
the period of the attracting orbit here? Indicate in which order the
points in the orbit occur.

**Up:** *Linear and Nonlinear Behavior*

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Created: Apr 3 1998 ---
Last modified: Wed Apr 8 19:26:13 1998