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
- \(\lambda(x_0)>0\)