To investigate the behavior of the population sequence for a in [1,4], we start by carrying out some numerical experiments. This is simplified by the use of a computer. In particular, two excellent applets written by Kanamaru of the University of Japan.
The page will take a few moments to load and display, while your computer prepares to run the applet. Be patient!
The applet shows two views of a population sequence. At the bottom is a time series view. At the top a web diagram view. Check you understand exactly what is being displayed (read further down the page for an explanation).
When first started, the applet has the parameter a (which is the same as our a) set to its largest possible value.
Task 3. Experimentally verify the following:
(1) Two sequences with seeds only slightly apart diverge very rapidly (exponentially). This is the mathematical definition of chaos.
(2) Each population sequence wanders all over [0,1]. More precisely, (xn) is dense in [0,1] - between any two elements of [0,1] there is a member of the sequence. This is the definition of ergodicicity.
Using the slider at the top of the applet, move a down to its
least possible value 2. You will see that the sequence (xn)
converges to 1/2 = 1-1/2. Slowly move the slider up to 3. For
each value of the parameter a, the sequence will converge to
1-1/a.
Now move the slider slowly, so that a is greater than 3. You will that a is greater than 3. You will notice that the sequence fails to converge. It oscillates between two limits! Since 3 is obviously an important number for the logistics map system, lets give it the name a1. Thus a1 marks the first bifurcation or period doubling of the logistics map.
Continue to increase a. You may see that around 3.45 there is another period-doubling. The sequence oscillates between 4 limits. But it is quite hard to pick this out, so we move on to a different applet, showing an alternative view of the behavior of the logistics map system.
The bifurcation applet is here:
Along the bottom is the parameter a (our a). The line above one particular a shows the values of xn, for n ³ 100. (Only taking n ³ 100 gives the sequence to settle down into its `usual' pattern.)
Thus we see that for 2 £ a £ 3, a single point we=symbol>£ 3, a single point which is the (single!) limit of the sequence. Then at 3 the diagram bifurcates, with the two visible points being those two numbers which the sequence is oscillating around. As mentioned above, at around 3.45, there is a second bifurcation, now clearly visible.
Write a2 for this second bifurcation point. And in general, let an be the nth bifurcation point. Let a¥ be the limit of the ans. Observe that a¥ is the smallest value of a where the population sequence (xn) is chaotic.
Looking at the diagram, it shows signs of being self similar - small pieces look like the whole picture. Lets pick out two numbers which capture the self similarity of the bifurcations. First let Dan = an+1 - an, and let dn = Dan / Dan+1. The limit of the dn exists, and is called d.
Second, look at the vertical line above a = a2. There are two points on this line. Let e1 be the distance between the two points. Then look at the line above a = a3. There are 4 points. Let e2 be the distance between the first two points. And so on ¼
Figure 1 is a simplified picture of this part of the bifurcation diagram, which make clear what are the an, Dan and en.
Task 4. As accurately as you can, determine a2 and
a3 etc, and a¥. And determine e1, e2
etc. [Perhaps you could write a program to do this better than
using the naked eye.]
Calculate dn. They should converge to about d = 4.67.
Calculate en. The ratio ei / ei+1 should converge to about e = 2.50.
After a¥ the diagram is quite confusing up until close to
4, where the population sequence is consistently chaotic. [Note
the white bands. Can you explain what is happeninsistently chaotic. [Note
the white bands. Can you explain what is happening for theses
values of a? Also note the dark lines. Any ideas what might cause
them? Can you detect any patterns in the oscillations?]
