I have been asked multiple times by various people of what chaotic dynamics are, the general
false assumption being that they are random (stochastic) systems. So I decided to write a quick
introduction of what *Chaos* actually is.

## Chaos

The term *chaos* stems from the systems experiencing irregularities based upon initial value conditions.
They are (usually) highly sensitive to initial value conditions, even a slight change as much as \(1\times10^{-12}\)
can lead to the system experiencing a completely different behavior after some time.
Chaotic systems are deterministic, meaning any value at any time of the system can be evaluated, they do not involve
random behavior.

A simple example of a chaotic system is the logistic map as defined as: $$ x_{n+1}=rx_n(1-x_n) $$ For some \(r, x_0\in\mathbb{R}\).

Let’s plot this with \(x_0=0.2\) and \(r=3.7\): Now let’s superimpose another plot of the same function but with \(x_0=0.20000001\): As can be examined the function behave completely different after some time \(t\). This is what’s called high sensitivity to initial values.

## Orbits

Considering a discrete time map \(x_{n+1}=f(x_n)\) with a given initial value \( x_{0} \), an orbit \(\{x_n\}\) refers to the set of points generated by the map starting at \( x_{0} \). This can be defined as, for any \(n\in\{k\in\mathbb{Z}|k\ge0\}\) $$ \{x_{n}\}=\{ x_0, f(x_0), f^2(x_0), …, f^n(x_0) \} $$ It is also quite often written in the shorthand \(\{x_n\}=\{f^{n}(x_0)\}_{n\ge0}\). This is also referred to as the orbit of \(x_0\) under \(f\).

### Fixed points

A fixed point satisfies the condition $$ f(x^{\star})=x^{\star} $$ such a point can be classified as follows:

- \(|f'(x^{\star})|<1\) then \(x^{\star}\) is a stable point
- \(|f'(x^{\star})|>1\) then \(x^{\star}\) is an unstable point
- \(|f'(x^{\star})|=0\) then \(x^{\star}\) is a super stable point

An example of a stable point is the logistic map with \(x_0=2\), \(r=2.8\): With those parameters the map converges on the point \(x^{\star}\approx 0.6428488290\) for which: $$ |f'(0.6428488290)|\approx0.8 $$

All initial values \(x_0\) that converge on a given stable point \(x^{\star}\) are called a basin of attraction.

### Period orbits

A period orbit that alternates between \(k\) fixed points \(\{x^{\star}_1, …, x^{\star}_k\}\) is called a period-*k* orbit.
All of these fixed points must be unique, that is \(\nexists m,n\) with \(m\ne n\) such that \(x^{\star}_m=x^{\star}_n\).

An example of a period-2 orbit is the logistic map with \(x_0=2\) and \(r=3.1\) as can be examined after a little time the map alternates between two points:

Period-*k* orbits can be classified with the determinants \(\Lambda_n\) of the jacobian matrix \(\textbf{J}\) of the function \(f\).
In a one-dimensional context this can be simplified to, for \(n\in\{n\in\mathbb{Z}|1\le n \le k\}\):

- \(|f'(x^{\star}_n)|<1\ \forall n\) the perid orbit is a sink
- \(|f'(x^{\star}_n)|>1\ \forall n\) the perid orbit is a source

## Lyapunov exponent

The Lyapunov exponent \(\lambda\) in relationship to an orbit \(\textbf{x}(t)\) with a nearby solution \(\textbf{x}(t)+\delta\textbf{x}(t)\) and with a small \(|\delta \textbf{x}(0)|\) is defined as: $$ |\delta\textbf{x}(t)| \simeq e^{\lambda t}|\delta\textbf{x}(0)| $$ For \(t>0\) and small \(|\delta \textbf{x}(0)|\). Now to obtain an expression for \(\lambda\) we can rearrange this to: $$ \lambda=\frac{1}{t}\ln\frac{|\delta\textbf{x}(t)|}{|\delta\textbf{x}(0)|} $$

The *maximal Lyapunov exponent* \(\lambda\) can be found by taking the limit for \(t\rightarrow\infty\) and for \(|\delta\textbf{x}|\rightarrow0\), for
\(\lambda>0\):

$$ \lambda=\lim_{t\rightarrow\infty}\lim_{|\delta\textbf{x}|\rightarrow0}\frac{1}{t}\ln\frac{|\delta\textbf{x}(t)|}{|\delta\textbf{x}(0)|} $$

For a system with discrete time orbit \(x_{n+1}=f(x_{n})\ \forall n\in\{k\in\mathbb{Z}|k\ge0\}\), this can be simplified as: $$ \lambda(x_0)=\lim_{n\rightarrow\infty}\frac{1}{n}\sum^{n-1}_{j=0}\ln|f'(x_j)| $$

## Quantifying Chaos

An orbit \(\{x_n\}_{n\ge0}\) of a map function \(f\) is considered *chaotic* if:

- \(\{x_n\}_{n\ge0}\in\mathbb{R}\ \forall n\)
- \(\{x_n\}_{n\ge0}\) is not periodic
- \(\lambda(x_0)>0\)