# Brief Introduction into Chaotic Dynamics

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
• $$\lambda(x_0)>0$$