Classification of periodic orbits

Section 2.5 Classification of periodic orbits

As mentioned right after the example on periodicty 2.6, a periodic point for \(f\) of period \(n\) is a fixed point of \(f^n\text{.}\) Treating the points of a periodic orbit this way allows us to extend the classification as fixed points to periodic orbits.

Definition 2.19.

Let \(f:\mathbb R \to \mathbb R\) be continuously differentiable and suppose that \(x_0\in\mathbb R\) is a periodic point of \(f\) with period \(n\text{.}\) Let \(F=f^n\text{.}\) We classify \(x_0\) and its orbit as

attractive, if \(|F'(x_0)| < 1\text{,}\)
super-attractive, if \(F'(x_0) =0 \text{,}\)
repulsive or repelling, if \(|F'(x_0)| > 1\text{,}\) or
neutral, if \(|F'(x_0)| = 1\text{,}\)

The number \(F'(x_0)\) is called the multiplier of the orbit. If, in the attractive case, the multiplier is zero, we say that the orbit is super-attractive.

There is a nice characterization of the multiplier of an orbit that allows us to compute it without explicitly computing a formula for \(f^n\text{.}\)

Lemma 2.20.

Suppose that

\begin{equation*} x_0 \to x_1 \to x_2 \to \cdots \to x_{n-1} \to x_0 \end{equation*}

is an orbit of period \(n\) for \(f:\mathbb R \to \mathbb R\text{.}\) Then the multiplier of the orbit is

\begin{equation*} f'(x_0)f'(x_1)\cdots f'(x_{n-1}). \end{equation*}

Proof.

First note that for an \(n=2\text{,}\) we can apply the chain rule to obtain

\begin{equation*} \frac{d}{dx}f^2(x) = \frac{d}{dx}f(f(x)) = f'(f(x))f'(x). \end{equation*}

Thus, if \(x_0 \to x_1 \to x_0\) is an orbit of period two and we evaluate that equation at \(x_0\text{,}\) we obtain

\begin{equation*} \left.\frac{d}{dx}f^2(x)\right|_{x=x_0} = f'(x_1)f'(x_0). \end{equation*}

The result for orbits longer than two can be proven by induction, since

\begin{equation*} \frac{d}{dx}f^n(x) = \frac{d}{dx}f(f^{n-1}(x)) = f'(f^{n-1}(x)) \frac{d}{dx}f^{n-1}(x). \end{equation*}

Note that the only way the product in Lemma 2.20 is zero, is if one of the terms is zero. This yields the following corollary.

Corollary 2.21.

A periodic orbit is super-attracting if and only if it contains a critical point.

Example 2.22.

Let \(f(x) = x^2-1\text{.}\) Note that \(f(0)=-1\) and \(f(-1)=0\) so that \(0\to1\to0\) forms an orbit of period 2. To see if this orbit is attractive, we examine

\begin{equation*} F(x) = f(f(x)) = (x^2-1)^2-1 = x^4-x^2. \end{equation*}

Note that \(F'(0)=0\) and \(F'(-1)=0\text{;}\) thus, the orbit is super-attractive.

The plots of \(f\) and \(f^2\text{,}\) together with \(y=x\text{,}\) are shown in Figure 2.23. Note that \(f\) has two fixed points shown in red. They can found by solving the equation \(x^2-1=x\) and they are both repulsive under iteration of \(f\text{.}\) The two super-attractive orbits of \(f^2\) are shown in green.

Figure 2.23. An attractive orbit of period two