Proof. For some c, d ∈ I we have f (c) = a, f (d) = b. So f (c) ¤ c, f (d) ≥ d. So

f (x) ’ x changes sign from c to d hence has a zero in between. QED.

Lemma 2.5.2 If J and K = [a, b] are compact intervals with K ‚ f (J) then

there is a compact subinterval L ‚ J such that f (L) = K.

Proof. Let c be the greatest point in J with f (c) = a. If f (x) = b for some

x > c, x ∈ J let d be the least. Then we may take L = [c, d]. If not, f (x) = b for

some x < c, x ∈ J. Let c be the largest. Let d be the the smallest x satisfying

x > c with f (x) = a. Notice that d ¤ c. We then take L = [c , d ]. QED

Notation. If I is a closed interval with end points a and b we write

I =< a, b >

when we do not want to specify which of the two end points is the larger.

Theorem 2.5.1 Sarkovsky Period three implies all periods.

Suppose that f has a 3-cycle

a ’ b ’ c ’ a ’ ··· .

Let a denote the leftmost of the three, and let us assume that

a < b < c.

(Reversing left and right (i.e. changing direction on the real line) and cycling

through the points makes this assumption harmless.) Let

I0 = [a, b], I1 = [b, c]

49

2.5. PERIOD 3 IMPLIES ALL PERIODS

so we have

f (I0 ) ⊃ I1 , f (I1 ) ⊃ I0 ∪ I1 .

By Lemma 2 the fact that f (I1 ) ⊃ I1 implies that there is a compact interval

A1 ‚ I1 with f (A1 ) = I1 . Since f (A1 ) = I1 ⊃ A1 there is a compact subinterval

A2 ‚ A1 with f (A2 ) = A1 . So

A2 ‚ A1 ‚ I, f —¦2 (A2 ) = I1 .

By induction proceed to ¬nd compact intervals with

An’2 ‚ An’3 ‚ · · · ‚ A2 ‚ A1 ‚ I1

with

f —¦(n’2) (An’2 ) = I1 .

Since f (I0 ) ⊃ I1 ⊃ An’2 there is an interval An’1 ‚ I0 with f (An’1 ) = An’2 .

Finally, since f (I1 ) ⊃ I0 there is a compact interval An ‚ I1 with f (An ) =

An’1 . So we have

An ’ An’1 ’ · · · ’ A1 ’ I1

where each interval maps onto the next and An ‚ I1 . By Lemma 1, f n has a

¬xed point, x, in An . But f (x) lies in I0 and all the higher iterates up to n lie

in I1 so the period can not be smaller than n. So there is a periodic point of

any period n ≥ 3.

Since f (I1 ) ⊃ I1 there is a ¬xed point in I1 , and since f (I0 ) ⊃ I1 , f (I1 ) ⊃ I0

there is a point of period two in I0 which is not a ¬xed point of f . QED

A more re¬ned analysis which we will omit shows that period 5 implies the

existence of all periods greater than 5 and period 2 and 4 (but not period 3). In

general any odd period implies the existence of periods of all higher order (and

all smaller even order). It is easy to graph the third iterate of the logistic map

to see that it crosses the diagonal for µ > 1 + (8). In fact, one can prove that

that at µ = 1 + (8) the graph of L—¦3 just touches the diagonal and strictly

µ

crosses it for µ > 1 + (8) = 3.8284 . . .. Hence in this range there are periodic

points of all periods.

50 CHAPTER 2. BIFURCATIONS

3.7 3.81

1 1

0.8 0.8

0.6 0.6

0.4 0.4

0.2 0.2

0 0

0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1

3.829 3.84

1 1

0.8 0.8

0.6 0.6

0.4 0.4

0.2 0.2

0 0

0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1

Figure 2.8: Plots of Lµ for µ = 3.7, 3.81, 3.83, 3.84

51

2.6. INTERMITTENCY.

2.6 Intermittency.

In this section we describe what happens to the period three orbits as we de-

√

crease µ √from slightly above the critical value 1 + 8 to slightly below it. For

µ = 1 + 8 + .002 the roots of P (x) ’ x where P := L—¦3 and the values of P

µ

at these roots are given by

x = roots ofP (x) ’ x P (x)

0 56.20068544683054

0.95756178779471 0.24278522730018

0.95516891475013 1.73457935568109

’6.13277919589328

0.73893250871724

0.52522791460709 1.73457935766594

0.50342728916956 0.24278522531345

0.16402371217410 1.73457935778151

0.15565787278717 0.24278522521922

We see that there is a stable period three orbit consisting of the points

0.1556 . . . , .5034 . . . , .9575 . . . .

If we choose our initial value of x close to .5034 . . . and plot the successive 199

iterates of Lµ applied to x we obtain the upper graph in Fig. 2.9. The lower

graph gives x(j + 3) ’ x(j) for j = 1 to 197. √

We will now decrease the parameter µ by .002 so that µ = 1 + 8 is the

parameter giving the onset of period three. For this value of the parameter, the

graph of P = L—¦3 just touches the line y = x at the three double roots of P (x)’x

µ

which are at 0.1599288 . . . , 0.514355 . . . ., 0.9563180 . . .. (Of course, the eighth

degree polynomial P (x) ’ x has two additional roots which correspond to the

two (unstable) ¬xed points of Lµ ; these are not of interest to us.) Since the

graph of P is tangent to the diagonal at the double roots, P (x) = 1 at these

points, so the period three orbit is not strictly speaking stable. But using the

same initial seed as above, we do get slow convergence to the period three orbit,

as is indicated by Fig. 2.10.

Most interesting is what happens just before the onset of the period three

cycle. Then P (x)’x has only two real roots corresponding to the ¬xed points of √

Lµ . The remaining six roots are complex. Nevertheless, if µ is close to 1+ 8 the √

e¬ects of these complex roots can be felt. In Fig. 2.11 we have taken µ = 1+ 8’

.002 and used the same initial seed x = .5034 and again plotted the successive

199 iterates of Lµ applied to x in the upper graph. Notice that there are portions

of this graph where the behavior is almost as if we were at a point of period three,

followed by some random looking behavior, then almost period three again and