**Up:** *Bifurcation*

# One-Dimensional Dynamical Systems

## Part 5: Bifurcation

#### Qualitative change of dynamics

The theorem of Hartman
and Grobman gives us the relationship between the slope of the graph
at a fixed point and whether this fixed point is attracting, repelling,
or neutral. For the Logistic map, we have already seen that the fixed
point other than 0 is attracting for
= 1.5, because
the slope is between -1 and 1. However, it is repelling for
= 3.1, since
the graph of the derivative map no longer has a slope between -1 and
1. Hence, as we increase
, the fixed
point changes from being attracting to being repelling. This scenario
is called a bifurcation.
- Recall that
*p* is a fixed point of *f* if *f*(*p*) =
*p*. Verify that the Logistic family
*f*(*x*) =
*x*
(1 - *x*) has fixed points *x* = 0 and *x* =
.

Compute the slope of the graph of the Logistic map at the fixed point
0 in terms of
. What is the
slope for the other fixed point?

- For what values of
are the fixed
points of the Logistic family attracting? Repelling? Neutral? Test
your values using the computer.

**Up:** *Bifurcation*

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Written by Hinke Osinga

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Created: Apr 6 1998 ---
Last modified: Thu Apr 9 14:18:16 1998