**Up:** *Bifurcation*

# One-Dimensional Dynamical Systems

## Part 5: Bifurcation

#### Bifurcation points

The Logistic map has an attracting fixed point for
= 1.5 that
becomes repelling as we increase
to 3.1.
**Definition (Bifurcation):**
If the qualitative dynamics of *f* changes as the parameter
is varied, we
call this a *bifurcation*.

**Definition (Bifurcation Point):**
In case of a bifurcation, there is a special value
_{b}
for which the following holds: the dynamics for
close to but
smaller than
_{b}
is qualitatively different from the dynamics for
close to but
larger than
_{b}
This value
_{b}
is called a *bifurcation point*.

Not every value of
is a
bifurcation point. Only when a qualitative change in the dynamics
occurs, we say that the function underwent a bifurcation.

- Approximately at what
value between
1.5 and 3.1 does a bifurcation occur?

- At exactly what
value between
1.5 and 3.1 does a bifurcation occur? Explain why you know this value
is exact.

In the following section we show how to visualize the dynamics as a
function of .

**Up:** *Bifurcation*

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Written by Hinke Osinga

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Created: Apr 6 1998 ---
Last modified: Wed Apr 8 19:48:50 1998