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 Figure 2.6.3. Again, we are dealing with the family of functions \(f_c(x) = x^2+c\text{.}\) 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 \(-0.75\text{,}\) it appears that we have an attracting orbit of period two. Why, exactly, does this happen?

First, let's explore the fixed points of \(f_c\text{;}\) we can find them by solving \(f_c(x)=x\text{:}\)

\begin{equation*} x^2+c = x \: \Longleftrightarrow \: x^2-x+c = 0. \end{equation*}

Applying the quadratic formula, we find

\begin{equation*} x = \frac{1\pm\sqrt{1-4c}}{2}. \end{equation*}

For \(c < 1/4\text{,}\) we have two real fixed points but a glance at the graphs from figure Figure 2.6.1 shows that it's the smaller of these two fixed points we're interested in. Of course, \(f'(x)=2x\text{,}\) so the value of the derivative at the smaller fixed point is \(1-\sqrt{1-4c}\text{.}\) Plugging \(c=-3/4\) into this formula, we find that this is \(-1\text{.}\) For \(c\) slightly larger than \(-3/4\text{,}\) this is bigger than \(-1\) and for \(c\) slightly smaller than \(-3/4\text{,}\) this is smaller than \(-1\text{.}\) 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 \(-3/4\text{.}\)

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

\begin{equation*} F_c(x) = f_c\circ f_c(x) = (x^2+c)^2+c = x^4 + 2cx^2+(c^2+c). \end{equation*}

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

\begin{equation} x^4 + 2cx^2+(c^2+c) = x \: \text{ or } \: x^4 + 2cx^2 - x + (c^2+c) = 0.\label{equation-f_squared}\tag{2.6.1} \end{equation}

Here is an observation that helps us factor this polynomial: Any point that is fixed by \(f_c\) must also be fixed by \(F_c\text{.}\) Thus, we expect \(x^2+c-x\) to be a factor of the polynomial in (2.6.1). Using this, we find that

\begin{equation*} x^4 + 2cx^2 - x + (c^2+c) = (x^2 - x + c)(x^2 + x + c + 1). \end{equation*}

We can then apply the quadratic formula to get the two new fixed points of \(F_c\text{,}\) namely

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

These two points form an orbit of period two for \(f_c\text{.}\) Since \(f_c'(x)=2x\) we can multiply those points by two and multiply the results to get the multiplier for the orbit. The result is:

\begin{equation*} \left(-1+\sqrt{-(3+4c)}\right)\left(-1-\sqrt{-(3+4c)}\right) = 4+4c. \end{equation*}

When \(c=-3/4\text{,}\) the multiplier is \(1\text{.}\) For \(c\) a little less than \(-3/4\text{,}\) the multiplier is a little less than one. Hence the orbit has become attractive.

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 \(c\text{.}\) This is shown in figure Figure 2.6.5 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 \(-0.75\text{.}\)

Figure 2.6.5 Bifurcation