Theorem2.5.2
If \(f:\mathbb C \to \mathbb C\) has an attractive or super-attractive orbit, then that orbit must attract at least one critical point.
Section2.5Parametrized families of functions
Rather than explore the behavior of a single function at a time, we can introduce a parameter and explore the range of behavior that arises in a whole family of functions. Two important examples are
- The quadratic family: \(f_c(x)=x^2+c\)
- The logistic family: \(f_{\lambda}(x)=\lambda x(1-x)\)
The cobweb plots shown back in Figure 2.2.1 are all chosen from the logistic family with \(\lambda=2.8\text{,}\) \(\lambda=3.2\text{,}\) and \(\lambda=4\text{.}\) Even in those three pictures with graphs that look so very similar, we see three different types of behavior: an attractive fixed point, an attractive orbit of period two, and chaos (which can be given a very technical meaning.
Figure 1 shows some cobweb plots for the quadratic family of functions. Note that the behavior we see is very similar to the behavior we see for the logistic family - a fact that will become more understandable once we study conjugacy in section Section 6
Subsection2.5.1The bifurcation diagram
A fabulous illustration of the types of behavior that can arise in a family of functions indexed by a single parameter and each with a single critical point can be generated as follows: For each value of the parameter, compute a large number points of the orbit of the critical point (maybe 1000 iterates). Since we're interested in long term behavior, rather than any transient behavior, discard the first few iterates (maybe 100). Then, plot the remaining points in a vertical column at the horizontal position indicated by the parameter.
The orbit of a critical point is called a critical orbit and its importance is due to the following theorem.
Regardless, the theorem has important implications for real iteration. For example, a polynomial of degree \(n\) can have at most \(n-1\) attractive orbits. Furthermore, if all the critical points happen to be real, we can find all the attractive behavior by simply iterating from the critical points. If we do this systematically for the quadratic family, plotting the columns to generate the bifurcation diagram, we get Figure 3
We can interpret this diagram as follows:
- For \(-0.75 < c < 0\text{,}\) there is an attractive fixed point.
- For \(-1.25 < c < -0.75\text{,}\) there is an attractive orbit of period \(2\text{.}\)
- As \(c\) passes from just above \(-0.75\) to just below \(-0.75\text{,}\) the dynamics of \(f_c\) undergo a bifurcation.
- For \(c\) just a little less than \(-1.25\text{,}\) there is an attractive orbit of period four. This orbit bifurcates soon into an attractive orbit of period 8. It appears that this behavior continues as \(c\) decreases.
- For \(c\) somewhere around \(c\approx -1.4\text{,}\) the period doubling appears to stop and we get more complicated behavior.
Generally, a bifurcation occurs at a parameter value \(c=c_0\) if the global dynamical behavior of the function \(f_c\) undergoes some qualitative change as \(c\) passes through \(c_0\text{.}\) There are number of different types of bifurcations that can occur, depending on the nature of the qualitative behavior under consideration. The bifurcations that are evident in Figure 3 in the range \(-1.4 < c < 0\) are called period doubling bifurcations.
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 \(-0.75\text{.}\)