One-Dimensional Dynamical Systems
Part 3: Iteration
is known as the Logistic map. The Logistic map depends on the parameter . As we already saw in the Introduction, the dynamics of the map depends on this parameter. However, the difference in the dynamics is not qualitatively significant for every different . In dynamical systems theory, we like to think of the whole set of maps depending on as one object. To express this idea, we speak of the one-parameter Logistic family. We discuss the dynamics of such a family by showing the qualitatively different types of possible dynamics, and by showing in which intervals of the parameter space these dynamics occur. The one-parameter Logistic family is our object of study.
The dependency on and x is nicely illustrated by the Java Applet Iteration of the Logistic Map of Andy Burbanks. Use this applet to study how the graph of the map changes as you increase , and what happens with the orbit of the chosen initial point. For fixed do the orbits of all the points in the interval [0,1] behave the same?
Written by Hinke Osinga
Comments to: firstname.lastname@example.org
Created: Apr 1 1998 --- Last modified: Wed Apr 8 18:44:56 1998