## Section2.5Classification of periodic orbits

As mentioned at the end of section one  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.

###### Definition2.5.1
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
1. attractive, if $|F'(x_0)| < 1\text{,}$
2. super-attractive, if $F'(x_0) =0 \text{,}$
3. repulsive or repelling, if $|F'(x_0)| > 1\text{,}$ or
4. 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 $f^n\text{.}$

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 2.5.2 is zero, is if one of the terms is zero. This yields the following corollary.

###### Example2.5.4

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 Figure 2.5.5. 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.