ν (0) = ’ (0, 0) < 0

‚P/‚x

since this time both numerator and denominator are negative. So the curve ν

has the same form as in the proof of the preceding proposition. This establishes

43

2.3. NEWTON™S METHOD AND FEIGENBAUM™S CONSTANT

the existence of the (strictly) period two points for µ < 0 and their absence for

µ > 0.

We now turn to the question of the stability of the ¬xed points and the period

two points. Condition (e) implies that »(µ) < ’1 for µ < 0 and »(µ) > ’1 for

µ > 0 so the ¬xed point is repelling to the left and attracting to the right of the

origin. As for the period two points, we wish to show that

‚F —¦2

(x, ν(x)) < 1

‚x

for x < 0. But (2.5) and ν (0) = 0 imply that imply that 0 is a critical point

for this function, and the value at this critical point is »(0)2 = 1. To complete

the proof we must show that this critical point is a local maximum. So we must

compute the second derivative at the origin. Calling this function φ we have

‚F —¦2

φ(x) := (x, ν(x))

‚x

‚ 2 F —¦2 ‚ 2 F —¦2

φ (x) = (x, ν(x)) + (x, ν(x))ν (x)

‚x2 ‚x‚µ

‚ 3 F —¦2 ‚ 3 F —¦2 ‚ 3 F —¦2

(x, ν(x))(ν (x))2 +

φ (x) = (x, ν(x)) + 2 2 (x, ν(x)ν (x)) +

3 2

‚x ‚x ‚µ ‚x‚µ

‚ 2 F —¦2

(x, ν(x))ν (x).

‚x‚µ

The middle two terms vanish at 0 since ν (0) = 0. The last term becomes

d»

(0)ν (0) < 0

dµ

by condition (e) and the fact that ν (0) < 0. In computing the third partial

derivative with respect to x of F —¦2 by the chain rule and by Leibniz™s rule, all

terms involving the second partial derivative vanish at (0, 0) by (2.5) and we

are left with

3

‚ 3F ‚F

2 3 (0, 0) (0, 0)

‚x ‚x

which is negative by assumptions (d) and (f). This completes the proof of the

proposition.

There are obvious variants on the theorem which involve changing signs in

hypotheses (e) and or (f). Thus we may have an attractive ¬xed point merging

with two repelling points of period two to produce a repelling ¬xed point, and/or

the direction of the bifurcation may be reversed.

2.3 Newton™s method and Feigenbaum™s constant

Although the values of bn for the logistic family are hard to compute except

by numerical experiment, the superattractive values can be found by applying

44 CHAPTER 2. BIFURCATIONS

Newton™s method to ¬nd the solution, sn , of the equation

1 1

n’1

—¦2

Lµ (x) = µx(1 ’ x).

Lµ ( )= , (2.10)

2 2

This is the equation for µ which says that 1 is a point of period 2n’1 of Lµ . Of

2

course we want to look for solutions for which 1 does not have lower period.

2

So we set

2n’1 1 1

P (µ) = Lµ ( ) ’

2 2

and apply the Newton algorithm

P (µ)

µk+1 = N (µk ), N (µ) = µ ’ .

P (µ)

with now denoting di¬erentiation with respect to µ. As a ¬rst step, must

compute P and P . For this we de¬ne the functions xk (µ) recursively by

1 1 1

, x1 (µ) = µ (1 ’ ),

x0 = xk+1 = Lµ (xk ),

2 2 2

so, we have

= [µxk (1 ’ xk ))]

xk+1

= xk (1 ’ xk ) + µxk (1 ’ xk ) ’ µxk xk

= xk (1 ’ xk ) + µ(1 ’ 2xk )xk .

Let

N = 2n’1

so that

1

P (µ) = xN ’ , P (µ) = xN (µ).

2

Thus, at each stage of the iteration in Newton™s method we compute P (µ)

and P (µ) by running the iteration scheme

=1

= µxk (1 ’ xk )

xk+1 x0 2

= xk (1 ’ xk ) + µ(1 ’ 2xk )xx

xk+1 x0 =0

for k = 0, . . . , N ’ 1. We substitute this into Newton™s method, get the next

value of µ, run the iteration to get the next value of P (µ) and P (µ) etc.

Suppose we have found s1 , s2 , ...., sn . What should we take as the initial

value of µ? De¬ne the numbers δn , n ≥ 2 recursively by δ2 = 4 and

sn’1 ’ sn’2

n ≥ 3.

δn = , (2.11)

sn ’ sn’1

We have already computed

√

s1 = 2, s2 = 1 + 5 = 3.23606797 . . . .

45

2.4. FEIGENBAUM RENORMALIZATION.

We take as our initial value in Newton™s method for ¬nding sn+1 the value

sn ’ sn’1

µn+1 = sn + .

δn

The following facts are observed:

For each n = 3, 4, . . . , 15, Newton™s method converges very rapidly, with no

changes in the ¬rst nineteen digits after six applications of Newton™s method for

¬nding s3 , after only one application of Newton™s method for s4 and s5 , and at

most four applications of Newton™s method for the computation of each of the

remaining values.

Suppose we stop our calculations for each sn when there is no further change

in the ¬rst 19 digits, and take the computed values as our sn . These values are

strictly increasing. In particular this implies that the sn we have computed do

1

not yield 2 as a point of lower period.

The sn approach a limiting value, 3.569945671205296863.

The δn approach a limiting value,

δ = 4.6692016148.

This value is known as Feigenbaum™s constant. While the limiting value of the

sn is particular to the logistic family, δ is “universal” in the sense that it applies

to a whole class of one dimensional iteration families. We shall go into this

point in the next section, where we will see that this is a renormalization group

phenomenon.