and (stable) points of period four appear. Initially these are stable, but as µ

increases they become unstable (at the value µ = 3.544090...) and bifurcate into

period eight points, initially stable.

2.1.7 Reprise.

The total scenario so far, as µ increases from 0 to about 3.55, is as follows:

For µ < b1 := 1, there is no non-zero ¬xed point. Past the ¬rst bifurcation

point, b1 = 1, the non-zero ¬xed point has appeared close to zero. When µ

reaches the ¬rst superattractive value , s1 := 2, the ¬xed point is at .5 and is

superattractive. As µ increases, the ¬xed point continues to move to the right.

Just after the second bifurcation point, b2 := 3, the ¬xed point has become

unstable and two stable points of period two appear, one to the right and one

to the left of .5. The leftmost period two point moves to the right as we increase

√

µ, and at µ = s2 := 1 + 5 = 3.23606797... the point .5 is a period two point,

and so the period two points are superattractive. When µ passes the second

√

bifurcation value b2 = 1 + 6 = 3.449.. the period two points have become

repelling and attracting period four points appear.

In fact, this scenario continues. The period 2n’1 points appear at bifurcation

values bn . They are initially attracting, and become superattracting at sn >

bn and become unstable past the next bifurcation value bn+1 > sn when the

period 2n points appear. The (numerically computed) bifurcation points and

superstable points are tabulated as

n bn sn

1 1.000000 2.000000

2 3.000000 3.236068

3 3.449499 3.498562

4 3.544090 3.554641

5 3.564407 3.566667

6 3.568759 3.569244

7 3.569692 3.569793

8 3.569891 3.569913

9 3.569934 3.569946

∞ 3.569946 3.569946

The values of the bn are obtained by numerical experiment. We shall describe

a method for computing the sn using Newton™s method. We should point out

35

2.2. LOCAL BIFURCATIONS.

that this is still just the beginning of the story. For example, an attractive

period three cycle appears at about 3.83. We shall come back to all of these

points, but ¬rst discuss theoretical problems associated to bifurcations.

2.2 Local bifurcations.

We will be studying the iteration (in x) of a function, F , of two real variables x

and µ . We will need to make various hypothesis concerning the di¬erentiability

of F . We will always assume usually it is at least C 2 (has continuous partial

derivatives up to the second order). We may also need C 3 in which case we

explicitly state this hypothesis. We write

Fµ (x) = F (x, µ)

and are interested in the change of behavior of Fµ as µ varies.

Before embarking on the study of bifurcations let us observe that if p is a

¬xed point of Fµ and Fµ (p) = 1, then for ν close to µ, the transformation Fν has

a unique ¬xed point close to p. Indeed, the implicit function theorem applies to

the function

P (x, ν) := F (x, ν) ’ x

since

‚P

(p, µ) = 0

‚x

by hypothesis. We conclude that there is a curve of ¬xed points x(ν) with

x(µ) = p.

2.2.1 The fold.

The ¬rst type of bifurcation we study is the fold bifurcation where there is no

(local) ¬xed point on one side of the bifurcation value, b, where a ¬xed point p

appears at µ = b with Fµ (p) = 1, and at the other side of b the map Fµ has two

¬xed points, one attracting and the other repelling.

As an example consider the quadratic family

Q(x, µ) = Qµ (x) := x2 + µ.

Fixed points must be solutions of the quadratic equation

x2 ’ x + µ = 0,

whose roots are

11

± 1 ’ 4µ.

p± =

22

For

1

µ>b=

4

36 CHAPTER 2. BIFURCATIONS

1.5 1.5 1

0.8

1 1

0.6

0.4

0.5 0.5

0.2

0 0 0

0 0.5 1 0 0.5 1 0 0.5 1

µ=.5 µ=.25 µ=0

Figure 2.5: y = x2 + µ for µ = .5, .25 and 0.

these roots are not real. The parabola x2 + µ lies entirely above the line y = x

and there are no ¬xed points.

At µ = 1 the parabola just touches the line y = x at the point ( 1 , 1 ) and so

4 22

1

p=

2

is a ¬xed point, with Qµ (p) = 2p = 1.

1

For µ < 4 the points p± are ¬xed points, with Qµ (p+ ) > 1 so it is repelling,

and Qµ (p’ ) < 1. We will have Qµ (p’ ) > ’1 so long as µ > ’ 3 , so on the

4

3 1

range ’ 4 < µ < 4 we have two ¬xed points, one repelling and one attracting.

We will now discuss the general phenomenon. In order not to clutter up the

notation, we assume that coordinates have been chosen so that b = 0 and p = 0.

So we make the standing assumption that p = 0 is a ¬xed point at µ = 0, i.e.

that

F (0, 0) = 0.

Proposition 2.2.1 (Fold bifurcation). Suppose that at the point (0, 0) we

have

‚2F

‚F ‚F

(a) (0, 0) = 1, (b) (0, 0) > 0, (c) (0, 0) > 0.

‚x2

‚x ‚µ

Then there are non-empty intervals (µ1 , 0) and (0, µ2 ) and > 0 so that

(i) If µ ∈ (µ1 , 0) then Fµ has two ¬xed points in (’ , ).

One is attracting and the other repelling.

(ii) If µ ∈ (0, µ2 ) then Fµ has no ¬xed points in (’ , ).

37

2.2. LOCAL BIFURCATIONS.

Proof. All the proofs in this section will be applications of the implicit function