📂Dynamics

Map System's Orbit

Definition¹

Let the smallest natural number satisfying $f^{k} (p) = p$ for maps $f : X \to X$ and $p \in X$ be $k \in \mathbb{N}$.

1. For map $f : X \to X$ and point $x \in X$, the set $\left\{ x , f(x) , f^{2} , \cdots \right\}$ under $f$ is called the orbit of $x$. Here, $x$ is called the initial value of the orbit.
2. An orbit $\left\{ p , f (p) , f^{2} (p) , \cdots \right\}$ with the initial value $p$ is called a Periodic-$k$ Orbit, and $p$ is called a Periodic-$k$ Point.
3. If $p$ is a sink of $f^{k}$, then its Periodic-$k$ Orbit is called a (Periodic) Sink, and if it’s a source of $f^{k}$, its Periodic-$k$ Orbit is called a (Periodic) Source.
4. If for some $N \in \mathbb{N}$ and all $n \ge N$, $f^{n+k} (p) = f^{n} (p)$ is satisfied, then $p$ is said to be Eventually Periodic.
5. If there exists a periodic orbit $\left\{ x_{1} , \cdots , x_{n} \right\}$ satisfying $\displaystyle \lim_{n \to \infty} | f^{n} (p) - x_{n} | = 0$ for orbit $\left\{ p , f (p) , f^{2} (p) , \cdots , f^{n} (p) , \cdots \right\}$, then $\left\{ p , f (p) , f^{2} (p) , \cdots , f^{n} (p) , \cdots \right\}$ is said to be Asymptotically Periodic.

Explanation

The existence of a Periodic-$k$ Orbit essentially means that $f^{k}$ has a fixed point. Thus, having a cycle or a fixed point simply becomes a matter of how many times the map is applied. Therefore, after conceptual study, all theorems and higher concepts are aligned to express based on the fixed point. Let’s consider ‘period’ as a generalization of ‘fixed point’ for natural numbers.

An orbit becoming exactly like its $\left\{ x_{1} , \cdots , x_{n} \right\}$ while being Asymptotically Periodic can also be considered Eventually Periodic. Moreover, an orbit converging to a Periodic Sink orbit is Asymptotically Periodic.

Meanwhile, for $X = \mathbb{R}$, one can consider the following simple theorem.

Theorem²

Let us call the Periodic-$k$ Orbit of $f$ as $\left\{ p_{1} , p_{2} , \cdots , p_{k} \right\}$.

If $\left| f '(p_{1}) \cdots f '(p_{k}) \right| < 1$, then $\left\{ p_{1} , p_{2} , \cdots , p_{k} \right\}$ is a sink, and if $\left| f '(p_{1}) \cdots f '(p_{k}) \right| > 1$, $\left\{ p_{1} , p_{2} , \cdots , p_{k} \right\}$ is a source.

Proof

By the chain rule,

$$ \begin{align*} ( f^{k} )' ( p_{1} ) =& \left( f \left( f^{k-1} \right) \right)' ( p_{1} ) \\ =& f ' \left( \left( f^{k-1} \right) \right) \left( f^{k-1} \right)' ( p_{1} ) \\ =& f ' \left( \left( f^{k-1} \right) \right) f ' \left( \left( f^{k-2} \right) \right) \cdots f ' ( p_{1} ) \\ =& f ' ( p_{k} ) f ' ( p_{k-1} ) \cdots f ' ( p_{1} ) \end{align*} $$

For a smooth $f : \mathbb{R} \to \mathbb{R}$, let’s say some $p \in \mathbb{R}$ is a fixed point.

[1] If $| f ' (p) | < 1$, then $p$ is a sink.

[2] If $| f ' (p) | > 1$, then $p$ is a source.

Applying The Criterion for Sinks and Sources in 1-Dimensional Maps to $| ( f^{k} )' ( p_{1} ) | = | f ' ( p_{k} ) f ' ( p_{k-1} ) \cdots f ' ( p_{1} ) |$ yields the desired result.

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