Lyapunov Exponents of One-Dimensional Maps
Definition1
Let’s assume an orbit $\left\{ x_{1} , x_{2} , x_{3} , \cdots \right\}$ of a smooth $1$-dimensional map $f : \mathbb{R} \to \mathbb{R}$ is given.
Lyapunov Number
The following is called the Lyapunov Number: $$L ( x_{1} ) : = \lim_{ n \to \infty } \left( \prod_{i = 1}^{n} | f ' (x_{i} ) | \right)^{1/n}$$
Lyapunov Exponent
The following is called the Lyapunov Exponent: $$h ( x_{1} ) := \lim_{n \to \infty } {{1} \over {n}} \sum_{i=1}^{n} \ln | f ' (x_{i} ) |$$ The Lyapunov Exponent is often also represented by $\lambda$.
Description
Intuitive Definition 2
The intuitive concept of the Lyapunov Number can be seen as ‘how much does a small difference in initial conditions affect the distant future’. Let’s denote the distance after a time of $n$ has passed from the initial condition $x_{1}$ and a very small distance $\delta_{1} \approx 0$ away at $x_{1} + \delta_{1}$ by $$ \delta_{n} := f^{n} \left( x_{1} + \delta_{1} \right) - f^{n} \left( x_{1} \right) $$ Here, when $\left| \delta_{n} \right| \approx \left| \delta_{1} \right| e^{n \lambda}$, $\lambda$ can be defined as the Lyapunov Exponent. If the Lyapunov Exponent is $0$ or greater, the higher the value, the greater the difference between $\left| \delta_{1} \right|$ and $\left| \delta_{n} \right|$, meaning the difference in initial conditions has a significant effect in the distant future. If it is less than $0$, the lower the value, the more similar the two values, meaning the initial condition differences have less impact on the distant future. If we further develop the formula for $\lambda$, by taking the logarithm on both sides, $$ \left| \delta_{n} \right| \approx \left| \delta_{1} \right| e^{n \lambda} \implies \ln \left| \delta_{n} \right| \approx \ln \left| \delta_{1} \right| n \lambda $$ and, $$ \begin{align*} \lambda \approx& {\frac{ 1 }{ n }} \ln \left| {\frac{ \delta_{n} }{ \delta_{1} }} \right| \\ =& {\frac{ 1 }{ n }} \ln \left| {\frac{ f^{n} \left( x_{1} + \delta_{1} \right) - f^{n} \left( x_{1} \right) }{ \delta_{1} }} \right| \\ \approx& {\frac{ 1 }{ n }} \ln \left| \left( f^{n} \right) ' \left( x_{1} \right) \right| \end{align*} $$ Meanwhile, $\left( f^{n} \right) ' \left( x_{1} \right)$, according to the chain rule, $$ \begin{align*} \left( f^{n} \right) ' \left( x_{1} \right) =& \left[ f \left( f^{n-1} \right) \left( x_{1} \right) \right] ' \\ =& \left( f^{n-1} \right) ' \left( x_{1} \right) \cdot f ' \left( f^{n-1} \left( x_{1} \right) \right) \\ =& \left( f^{n-1} \right) ' \left( x_{1} \right) f ' \left( x_{n} \right) \\ =& \left( f^{n-2} \right) ' \left( x_{1} \right) \cdot f ' \left( f^{n-2} \left( x_{1} \right) \right) f ' \left( x_{n} \right) \\ =& \left( f^{n-2} \right) ' \left( x_{1} \right) \cdot f ' \left( x_{n-1} \right) f ' \left( x_{n} \right) \\ & \vdots \\ =& \prod_{k=1}^{n} f ' \left( x_{k} \right) \end{align*} $$ Therefore, $$ \begin{align*} \lambda \approx& {\frac{ 1 }{ n }} \ln \left| \left( f^{n} \right) ' \left( x_{1} \right) \right| \\ =& {\frac{ 1 }{ n }} \ln \left| \prod_{k=1}^{n} f ' \left( x_{k} \right) \right| \\ =& {\frac{ 1 }{ n }} \sum_{k=1}^{n} \ln \left| f ' \left( x_{k} \right) \right| \end{align*} $$ thus, for large enough $n \in \mathbb{N}$, the following approximation can be induced. $$ \lambda \approx {{1} \over {n}} \sum_{k=1}^{n} \ln | f ' (x_{k} ) | \approx \lim_{N \to \infty} {{1} \over {N}} \sum_{k=1}^{N} \ln | f ' (x_{k} ) | = h \left( x_{1} \right) $$
Chaos Theory
Thinking again about the concepts of sinks and sources, a sink can be seen as a ‘convergence point’ where nearby points gather, and a source as a ‘divergence point’ where previously close points start moving apart. This can also be extended similarly for periodic orbits.
Sink and Source Judgment for Orbits: > Let’s 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 $\left| f '(p_{1}) \cdots f '(p_{k}) \right| > 1$, then $\left\{ p_{1} , p_{2} , \cdots , p_{k} \right\}$ is a source.
The Lyapunov Number was introduced to extend these concepts beyond periodic orbits. When considering sinks as showing a stable tendency and sources as indicating a spreading fluctuation, if the infinite product of derivatives represented by $L ( x_{1} )$ is greater than $1$, it implicitly indicates that the orbit of $x_{1}$ is a source. From this viewpoint, the Lyapunov Exponent defines the concept of chaos along with asymptotically periodic, or provides the following theorem.
Theorem3
Let’s assume that $\left\{ x_{1} , x_{2} , \cdots \right\}$, satisfying $f ' (x_{i} ) \ne 0$, is asymptotically periodic to the periodic-$k$ orbit $\left\{ y_{1} , y_{2} , \cdots , y_{k} , \cdots \right\}$ in the smooth map $f : \mathbb{R} \to \mathbb{R}$. Then, both orbits have the same Lyapunov Exponent.
Proof
Part 1. The average of a sequence converges to the limit of the original sequence.
As a simple fact, if $\displaystyle \lim_{n \to \infty} s_{n} = s$, then for $m \in \mathbb{Z}$, $\displaystyle \lim_{n \to \infty} s_{n+m} = s$ holds.
Then, the average also converges to $s$, and mathematically, $\displaystyle \lim_{n \to \infty} {{1} \over {n}} \sum_{i=1}^{n} s_{i} = s$
Part 2.
If $k=1$, then $y_{1}$ becomes the fixed point of $f$. $\displaystyle \lim_{ n \to \infty } x_{n} = y_{1}$ and since $f$ is smooth,
$$ \lim_{n \to \infty } f ' (x_{n}) = f ' \left( \lim_{n \to \infty} x_{n} \right) = f ' ( y_{1} ) $$
Meanwhile, $\ln | \cdot | $ is also a continuous function, so
$$ \lim_{n \to \infty } \ln | f ' (x_{n}) | = \ln \left| \lim_{n \to \infty} f ' ( x_{n} ) \right| = \ln | f ' ( y_{1} ) | $$
Therefore, by Part 1.,
$$ \begin{align*} h ( x_{1} ) =& \lim_{n \to \infty } {{1} \over {n} } \sum_{i=1}^{n} \ln | f ' (x_{n}) | \\ =& \lim_{n \to \infty } \ln | f ' (x_{n}) | \\ =& {{1} \over {1}} \ln | f ' (y_{1} ) | \\ =& h (y_{1} ) \end{align*} $$
Part 3.
Assuming under $f$ the Lyapunov Number of $x_{1}$ is $\displaystyle L := \lim_{ n \to \infty } \left( | f ' ( x_{1} ) | \cdots | f ' ( x_{n} ) | \right)^{1/n}$. By the chain rule, for $i = 1, \cdots , k$, $( f^{k} )' ( x_{i} ) = f ' ( x_{1} ) \cdots f ' ( x_{k} )$, therefore, under $f^{k}$ the Lyapunov Number of $x_{1}$ is
$$ \begin{align*} & \lim_{ n \to \infty } \left( \left| (f^{k})' ( x_{1} ) \right| \cdots \left| (f^{k}) ’ ( x_{n} ) \right| \right)^{1/n} \\ =& \lim_{ n \to \infty } \left( | f ' ( x_{1} ) | \cdots | f ' ( x_{n} ) | \right)^{k/n} \\ =& L^{k} \end{align*} $$
The calculation process naturally implies the reverse is also true, and under $f$ the Lyapunov Exponent of $x_{1}$ is the same as under $ h = f^{k}$.
Part 4.
If $k > 1$, then $y_{1}$ is the fixed point of $f^{k}$ and $\left\{ x_{1} , x_{2} , \cdots \right\}$ is asymptotically periodic to $\left\{ y_{1} , y_{2} , \cdots , y_{k}, \cdots \right\}$.
$$ \begin{align*} h(x_{1} ) =& \lim_{ n \to \infty } {{1} \over {n}} \left( \ln | f ' ( x_{1} ) | + \cdots + \ln | f ' ( x_{n} ) | \right) \\ =& \lim_{ n \to \infty } {{1} \over {n}} \ln \left( | f ' ( x_{1} ) | \cdots | f ' ( x_{n} ) | \right) \\ =& \lim_{ n \to \infty } {{1} \over {k \cdot n}} \ln \left( | f ' ( x_{1} ) |^{k} \cdots | f ' ( x_{n} ) |^{k} \right) \\ =& {{1} \over {k}} \lim_{ n \to \infty } \ln \left( | (f^{k} ) ’ ( x_{n} ) | \right) = {{1} \over {k}} \ln \left| ( f^{k} )' ( y_{1} ) \right| \end{align*} $$
By Part 2., under $f^{k}$ the Lyapunov Exponent of $x_{1}$ is
$$ \ln \left| (f^{k})' (y_{1}) \right| $$
By Part 3., under $f$ the Lyapunov Exponent of $x_{1}$ is
$$ h( x_{1} ) = {{1} \over {k}} \ln \left| ( f^{k} )' ( y_{1} ) \right| = h ( y_{1} ) $$
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