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

# One-Dimensional Dynamical Systems

## Part 4: Linear and Nonlinear Behavior

#### Hartman and Grobman

Locally near a point *x*, a differential function can be
approximated by its derivative at *x*. That is, the graph of
the function locally near *x* is almost equal to a straight
line with a slope equal to the derivative at *x*. Hence, the
derivative *Df*(*x*) at a specific point *x* can
be interpreted as a (linear) map, that has this straight line as its
graph. This map is called the derivative map. However, if the
functions are almost the same near *x*, what about the behavior
of the two dynamical systems?
**Theorem by Hartman(1960) and Grobman (1962):**
For any fixed point *p* of a function *f* the following
is true in a small neighborhood of *p*: the behavior
of the dynamical system given by *f* is qualitatively the same
as the behavior of the linear dynamical system given by
*L*(*x*) = *Df*(*p*) *x*.

Note that the theorem is only true on a small neighborhood of the
fixed point *p*. In particular if other fixed points or
periodic points are close to *p*, this neighborhood is very
small! You can now find a relationship between the slope of the graph
at the fixed point and whether the fixed point is attracting,
repelling, or neutral.

*The fixed point other than 0 is attracting for
= 1.5.*

Recall your investigations in the
Warm Up Questions. In the
figure above (right) the graph of the derivative map at the fixed
point is drawn in read. Also drawn are the two special lines with
slope 1 and -1. The figure clearly shows that the derivative at the
fixed point lies in between -1 and 1, so that the fixed point is
attracting.

*The fixed point other than 0 is repelling for
= 3.1.*

For = 3.1 the
graph of the derivative map no longer has a slope between -1 and
1. Hence, the fixed point is now repelling.

- What are the slopes of the graph of the Logistic map at the fixed
points for = 1.5
and 3.1?

*The fixed point other than 0 is attracting for
= 2.5.*

- Compare the figure for
= 2.5 with
the figure for
= 1.5. Notice
the difference in behavior of the graphical iterations. Why do you
think this difference is there? (Hint: look at the slope of the graph
at the attracting fixed point.)

- Recall that in order to find period-2 points, you graphed the
second iterate of the map. Can you find a relationship between the
slope of the graph of the second iterate of the function and whether the
period-2 points are attracting, repelling, or neutral?

- Describe geometrically how your relationship might generalize
to period-
*n* points.

Note: The procedure you found only works for periodic points with
small periods. Although it is nice to have the condition above for
whether a period-*n* point is attracting, it becomes
computationally difficult to accurately graph the *n*-th iterate
of a map as *n* becomes large.

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

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Created: Apr 3 1998 ---
Last modified: Thu Apr 9 14:02:09 1998