2.2 The Logistic Map in a spreadsheet

The Logistic Map takes the following functional form:

\[\begin{equation} Y_{t+1} = r*Y_t*(1-Y_t) \tag{2.2} \end{equation}\]

To get started, copy the spreadsheet from the previous assignment to a new sheet. The parameters are the same as for the Linear Map, there has to be an initial value \(Y_{t=0}\) (no longer explicit as a constant in equation (2.2)) and the control parameter \(r\). What will have to change is

  • Start with the following values for control parameter \(r\):
    • \(r = 1.9\)
    • \(Y_0 = 0.01\) (in A6).
  • Take good notice of what is constant (parameter \(r\)), so for which the $ must be used, and what must change on every iterative step (variable \(Y_t\)).

2.2.1 Visualizing the time series and explore its behaviour

  • Create the time series graphs as for the Linear Map.

To study the behavior of the Logistic Map you can start playing around with the values for the parameters and the initial values in cells B5 and B6.

  • Be sure to try the following settings for \(r\):
    • \(r = 0.9\)
    • \(r = 1.9\)
    • \(r = 2.9\)
    • \(r = 3.3\)
    • \(r = 3.52\)
    • \(r = 3.9\)
  • Set \(r\) at \(4.0\):
    • Repeat the iterative process from A10 to A310 (300 steps)
    • Now copy A10:A310 to B9:B309 (i.e., move it one cell to the right, and one cell up)
    • Select both columns (A10 to B309!) and make a scatter-plot

2.2.2 The return plot

The plot you just produced is a so called return plot, in which you have plotted \(Y_{t+1}\) against \(Y_t\).

  • Can you explain the pattern you see (a ‘parabola’) by looking closely at the functional form of the Logistic Map? (hint: it’s also called Quadratic Map)
    • Look at what happens in the return plot when you change the value of the parameter \(r\) (in A5).
    • What do you expect the return plot of the Linear Map will look like? Try it!

The meaning and use of this plot was discussed in the next session