One-Dimensional Dynamical Systems
Part 4: Linear and Nonlinear 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.
Written by Hinke Osinga
Comments to: firstname.lastname@example.org
Created: Apr 3 1998 --- Last modified: Wed Apr 8 19:26:13 1998