Subsection2.5.2The period doubling cascade
Let's work towards a deeper, theoretical understanding of the period doubling that we see in the bifurcation diagram of Figure 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 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. \tag{2.5.1} \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 (2.5.1). 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*} (-1+\sqrt{-(3+4c)})(-1-\sqrt{-(3+4c)}) = 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 4 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 \(-0.75\text{.}\)