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.

When you load the Package, the Maple package plots is loaded automatically, and so is the file Logistic.mpl, so that the current function is set to the Logistic map

f(x) = x (1 - x).

Function Iteration with = 2.

Instructions to Iterate

• Upon loading the Package, there are some messages printed. The last line says

  lambda = 2.000000, x0 = .940000

  These are the variables and parameters that you can change. If you type

  lambda := 3.1;

  the parameter is set to 3.1. Similarly x0 is the initial value for which the orbit is computed (x0 should be chosen in the closed interval [0,1].)
  
  

• The procedures

  Iterates(20);
PlotIterates(20);

  Produce 20 iterates of the initial condition x0. Iterates produces a list of numbers and PlotIterates produces a plot of the iterates versus time. Of course, you can choose any number instead of 20.
  
  

• Experiment with different values for and x₀.

Instructions for Maple: Graphical Iteration

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.

For the Logistic map, pick a value between 0 and 4, and iterate as follows:

Visualising an orbit with graphical iteration for the Logistic map.

Instructions to Iterate

• You can change the variable x0 and the parameter lambda by typing something like

  x0 := 0.12345;

• Next, type

  PlotGraphical(20)

  A figure will appear with the graph of the Logistic map for this specific value. The line y = x is drawn as well, and you see the graphical iteration of x0 up to 20 iterates with the last iterates drawn in red.