P (x, µ) := F (x, µ) ’ x.

Then by our standing hypothesis we have

P (0, 0) = 0

and condition (c) gives

‚P

(0, 0) > 0.

‚µ

The implicit function theorem gives a unique function µ(x) with µ(0) = 0 and

P (x, µ(x)) ≡ 0.

The formula for the derivative in the implicit function theorem gives

‚P/‚x

µ (x) = ’

‚P/‚µ

which vanishes at the origin by assumption (a). We then may compute the

second derivative, µ , via the chain rule; using the fact that µ (0) = 0 we obtain

‚ 2 P/‚x2

µ (0) = ’ (0, 0).

‚P/‚µ

This is negative by assumptions (b) and (c). In other words,

µ (0) = 0, and µ (0) < 0

so µ(x) has a maximum at x = 0, and this maximum value is 0. In the (x, µ)

plane, the graph of µ(x) looks locally like a parabola pointing in the lower half

plane with its apex at the origin. If we rotate this picture clockwise by ninety

degrees, this says that there are no points on this curve sitting over positive µ

values, i.e. no ¬xed points for positive µ, and two ¬xed points for µ < 0.

Now consider the function ‚F (x, µ(x)). The derivative of this function with

‚x

respect to x is

‚2F ‚2F

(x, µ(x)) + (x, µ(x))µ (x).

‚x2 ‚x‚µ

By assumption (b) and µ (0) = 0, this expression is positive at x = 0 and

so ‚F (x, µ(x)) is an increasing function in a neighborhood of the origin while

‚x

‚F

‚x (0, 0) = 1. But this says that

Fµ (x) < 1

on the lower ¬xed point and

Fµ (x) > 1

at the upper ¬xed point, completing the proof of the proposition. We should

point out that changing the sign in (b) or (c) interchanges the role of the two

intervals.

38 CHAPTER 2. BIFURCATIONS

2.2.2 Period doubling.

We now turn to the period doubling bifurcation. This is what happens when we

pass through a bifurcation value with

‚F

(0, 0) = ’1. (2.3)

‚x

We saw examples in the preceding section.

To visualize the phenomenon we plot the function L—¦2 for the values µ = 2.9

µ

and µ = 3.3 in Figure 2.6. For µ = 2.9 the curve crosses the diagonal at a

single point, which is in fact a ¬xed point of Lµ and hence of L—¦2 . This ¬xed

µ

point is stable. For µ = 3.3 there are three crossings. The ¬xed point of Lµ has

derivative smaller than ’1, and hence the corresponding ¬xed point of L—¦2 has

µ

derivative greater than one. The two other crossings correspond to the stable

period two orbit.

We now turn to the general theory: Notice that the partial derivative of

F (x, µ) ’ x with respect to x is ’2 at the origin. In particular it does not

vanish, so we can now solve for x as a function of µ; there is a unique branch of

¬xed points, x(µ), passing through the origin. Let »(µ) denote the derivative

of Fµ with respect to x at the ¬xed point, x(µ), i.e. de¬ne

‚F

»(µ) := (x(µ), µ).

‚x

As notation, let us set

—¦2

Fµ := Fµ —¦ Fµ

and de¬ne

F —¦2 (x, µ) := Fµ (x).

—¦2

Notice that

—¦2

(Fµ ) (x) = Fµ (Fµ (x))Fµ (x)

by the chain rule so

(F0 ) (0) = (F0 (0))2 = 1.

—¦2

Hence

(Fµ ) (x) = Fµ (Fµ (x))Fµ (x)2 + Fµ (Fµ (x))Fµ (x)

—¦2

(2.4)

which vanishes at x = 0, µ = 0. In other words,

‚ 2 F —¦2

(0, 0) = 0. (2.5)

‚x2

Let us absorb the import of this equation. One might think that if we set

—¦2

Gµ = Fµ , then Gµ (0) = 1, so all we need to do is apply Proposition 1 to Gµ .

But (2.5) shows that the key condition (b) of Proposition 1 is violated, and

hence we must make some alternative hypotheses. The hypotheses that we will

make will involve the second and the third partial derivatives of F , and also

d»

that »(µ) really passes through ’1, i.e. dµ (0) = 0.

39

2.2. LOCAL BIFURCATIONS.

1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Figure 2.6: Plots of L—¦2 for µ = 2.9 (dotted curve) and µ = 3.3.

µ

40 CHAPTER 2. BIFURCATIONS

To understand the hypothesis about involving the partial derivatives of F ,

let us di¬erentiate (2.4) once more with respect to x to obtain