One-Dimensional Dynamical Systems

First make sure that you have a personal copy of the package Chaos.mpl and that you know how to start maple and use this package.

Instructions to graph higher iterates

• Pick a value for and type lambda := .
  
  
• Type PlotGraphical to get the graph of the corresponding function. If you do not want to see the graphical iteration, you can also type

  ShowGraph(1, colour);

  where colour is the color of the graph. You can choose standard colours (red, blue, green, yellow, orange, etcetera), or use COLOR(RGB, r, g, b), with r, g, b in the interval [0,1].
  
  

• To add a graph of the second iterate to this picture, type

  AddGraph(%, 2, other-color);

  You get a new picture that contains both graphs.
  
  

Instructions for Maple: Orbit Diagram

First make sure that you have a personal copy of the package Chaos.mpl and that you know how to start maple and use this package.

An enlargement of the orbit diagram for the Logistic family.

How to Study the Orbit Diagram

• To start the computation, type

  BifDiagram(first-its, total-its);

  The program begins by computing total-its iterates of an arbitrary point x₀. However, iterates are displayed only after first-its iterates. Therefore, the transients are not shown in the picture. In Maple it is not possible to have the picture being partially drawn already while computing. Therefore, it may take quite a while. In total 200 orbits are computed for each picture, and a count-down is being printed so that you can guess how long it takes.
  
  

• To get a picture that reflects the eventual attracting behavior more accurately, increase first-its and total-its.
  
  
• You can zoom in with

  display(%, view=[3.1..3.8, 0.6..0.8]);

  However, when you zoom in for , you will only see a selection of the 200 values used for the computation of the bigger picture. If you redo the computation with

  BifDiagram(first-its, total-its, [3.1, 3.8]);

  the 200 values will be taken from this smaller interval. If you zoom in for x0 you probably want to increase first-its and total-its even more. Therefore, you may want to type

  BifDiagram(first-its, total-its, [3.1, 3.8], [0.6, 0.8]);

  The computations are done now only for the ranges specified. (So you cannot zoom out on this picture!)