The period doubling cascade

Subsection 2.6.2 The period doubling cascade

Let's work towards a deeper, theoretical understanding of the period doubling that we see in the bifurcation diagram of Figure 2.26. Again, we are dealing with the family of functions

f_{c} (x) = x^{2} + c .

For

c

just a bit larger than

- 0.75

it appears that we have an attractive fixed point while, for

c

just a bit smaller than

- 0.75,

it appears that we have an attracting orbit of period two. Why, exactly does this happen?

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First, let's explore the fixed points of

f_{c};

we can find them by solving

f_{c} (x) = x :

x^{2} + c = x ⟺ x^{2} - x + c = 0.

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Applying the quadratic formula, we find

x = \frac{1 \pm \sqrt{1 - 4 c}}{2} .

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For

c < 1 / 4,

we have two real fixed points but a glance at the graphs from Figure 2.24 shows that it's the smaller of these two fixed points we're interested in. Of course,

f^{'} (x) = 2 x,

so the value of the derivative at the smaller fixed point is

1 - \sqrt{1 - 4 c} .

Plugging

c = - 3 / 4

into this formula, we find that this is

- 1 .

For

c

slightly larger than

- 3 / 4,

this is bigger than

- 1

and for

c

slightly smaller than

- 3 / 4,

this is smaller than

- 1 .

This explains why we have an attractive fixed point for

c

slightly larger than

- 3 / 4

that is no longer attractive once

c

passes below

- 3 / 4 .

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Now, we ask - why does the attractive orbit of period two appear as the attractive fixed point disappears? To see this, we consider the function

F_{c} (x) = f_{c} \circ f_{c} (x) = (x^{2} + c)^{2} + c = x^{4} + 2 c x^{2} + (c^{2} + c) .

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We are interested in the fixed points, thus we must solve

\begin{matrix} (2.1) & x^{4} + 2 c x^{2} + (c^{2} + c) = x or x^{4} + 2 c x^{2} - x + (c^{2} + c) = 0. \end{matrix}

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Here is an observation that helps us factor this polynomial: Any point that is fixed by

f_{c}

must also be fixed by

F_{c} .

Thus, we expect

x^{2} + c - x

to be a factor of the polynomial in (2.1). Using this, we find that

x^{4} + 2 c x^{2} - x + (c^{2} + c) = (x^{2} - x + c) (x^{2} + x + c + 1) .

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We can then apply the quadratic formula to get the two new fixed points of

F_{c},

namely

x = \frac{- 1 \pm \sqrt{1 - 4 (c + 1)}}{2} = \frac{- 1 \pm \sqrt{- (3 + 4 c)}}{2} .

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These two points form an orbit of period two for

f_{c} .

Since

f_{c}^{'} (x) = 2 x

we can multiply those points by two and multiply the results to get the multiplier for the orbit. The result is:

(- 1 + \sqrt{- (3 + 4 c)}) (- 1 - \sqrt{- (3 + 4 c)}) = 4 + 4 c .

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When

c = - 3 / 4,

the multiplier is

1 .

For

c

a little less than

- 3 / 4,

the multiplier is a little less than one. Hence the orbit has become attractive.

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A nice way to visualize this is to plot

f_{c}^{2}

together with

f_{c}

and

y = x

on the same set of axes for a few different choices of

c .

This is shown in Figure 2.27 where we can see exactly how The fixed point went from attractive to repulsive while an attractive orbit of period two showed up as

c

passed below

- 0.75 .

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Figure 2.27. Bifurcation

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Note that our cobweb tool allows you to plot iterates of

f

on the plot to assist in this type of analysis.

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Checkpoint 2.28.

Consider the logistic family \(f_{\lambda}(x) = \lambda x(1-x)\text{.}\)

Using our cobweb tool