As accurately as you can, determine a₂, a₃ etc, and a_¥. And determine e₁, e₂ etc. Calculate the d_n. Do they converge to about 4.67? The ratio e_i /e_i+1 should converge to about 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 happening for theses values of a? Also note the dark lines. Any ideas what might cause them? Can you detect aas what might cause them? Can you detect any patterns in the oscillations?]

This task is simplified with the use of this applet, created by Jarrod Pickens:

Here are values obtained by using the applet:

a₁ = 3
a₂ = 3.4494
a₃ = 3.5441
a₄ = 3.5644
a₅ = 3.56875
a₆ = 3.5697
a_¥ = 3.5699



e₁ = 0.406
e₂ = 0.157
e₃ = 0.062
e₄ = 0.0247
e₅ = 0.00987



The values above give estimates of 4.66 for d and 2.51 for e.

The white bands represent areas of periodic behavior even after a_¥. Looking carefully one can find cycles of period 6 and period 5. The dark lines show `favored' cycle points. These are the limit points of the sequences (f²ⁿ (x₁)), (f³ⁿ(x₁)) etc.