Applications - A Guide to MATLAB for Beginners and Experienced Users - page 164

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

0 10 20 30 40 50 60 70 80 90 100

(4) If 3 3.56994..., Then There Is a PeriodicCycle

The theory is quite subtle. For a fuller explanation, the reader may consult
Encounters with Chaos , by Denny Gulick, McGraw-Hill, 1992, Section 1.5. In
fact there is a sequence

< u <

√ 6 < u 2 < u 3 <...< 4 ,

u 0 = 3 < u 1 = 1 +

suchthat between u 0 and u 1 there is a cycle of period 2, between u 1 and u 2
there is a cycle of period 4, and in general, between u k and u k + 1 there is a
cycle of period 2 k + 1 . One als o know s that, at least for small k , one has the
approximation u k + 1 ≈ 1 +

3 + u k .So

u1=1+sqrt(6)

u1 =
3.44948974278318

u2approx = 1 + sqrt(3 + u1)

u2approx =
3.53958456106175

This explains the oscillatory behavior we saw in the last of the original four
examples (with u 0 < u = 3 . 4 < u 1 ). Here is the behavior for u 1 < u = 3 . 5 < u 2 .
The command bar is particularly effective here for spotting the cycle of
order 4.

X = itseq(f, 0.75, 100, 3.5);
bar(X); axis([0 100 0 0.9])