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Subsection 3.7.3 The period two bulb

After the main cardioid, the next largest component is the disk attached to the main cardioid at the point \(c=-3/4\text{.}\) This is, in fact, an actual disk attached at that exact point. We can prove this in much the same way that we derived the formula for the main cardioid, though the algebra is a little more involved.

This disk is known as the period two bulb; every function \(f_c\) where \(c\) is chosen from the period two bulb has an attractive orbit of period two. We can characterize this algebraically by first defining \(F_c(z)=f_c(f_c(z))\text{.}\) A point is then part of an attractive orbit of period 2 for \(f_c\) if

\begin{align*} F_c(z) &= z\\ |F_c'(z)| &\lt 1. \end{align*}

Mimicking the construction of the main cardioid in the period 1 case, we find that \(z\) and \(c\) must satisfy

\begin{align*} F_c(z) &= z\\ F_c'(z) &= e^{2\pi i t}, \end{align*}

for some \(t\text{.}\) Taking into account the fact that \(F_c(z) = (z^2+c)^2+c\text{,}\) we get

\begin{align*} (z^2+c)^2+c &= z\\ 4z(z^2+c) &= e^{2\pi i t}. \end{align*}

While clearly this is a bit more involved than the period one case, a computer algebra system makes quick work of it. Listing Listing 3.7.5 shows how to solve this system with Mathematica and listing Listing 3.7.6 shows how to solve this system with Sage.

f[c_][z_] = z^2 + c;
F[c_][z_] = Nest[f[c], z, 2];
eqs = {F[c][z] == z, F[c]'[z] == Exp[2 Pi*I*t]};
Expand[c /. Last[Solve[eqs, {c, z}]]]
(* Out:
	-1 + E^((2*I)*Pi*t)/4
*)
Listing 3.7.5 Mathematica code to parametrize the period two bulb
Listing 3.7.6 Sage code to parametrize the period two bulb

Note that both program listings Listing 3.7.5 and Listing 3.7.6 yield the result

\begin{equation*} c=-1 + \frac{1}{4}e^{2\pi i t}, \end{equation*}

which parametrizes a circle of radius \(1/4\) centered at the point \(-1\text{.}\)

Also, it's certainly feasible to solve this system by hand. The expression \(F_c(z)-z\) factors so that the system really just involves a couple of quadratics. Use of the computer will be quite convenient as we move into higher order examples, though.

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