Chaos in stb

C library that simulates fractals and attractors (only in stb). You can generate them into terminal as ASCII art or into image (using stb library).

Table of contents

• Bifurcation diagram
• Attractors
  • Lorenz attractor
  • Thomas attractor
  • Rössler attractor
  • Chen & Lu Chen attractor
  • Rabinovich–Fabrikant attractor
  • Hénon map
• Fractals
  • Cantor set
  • Sierpiński carpet
  • Sierpiński triangle
  • Mandelbrot set
  • Koch snowflake
  • Barnsley fern

Bifurcation diagram

• Available in ASCII
• Available in stb

$$x_{n+1} = x_n \cdot k \cdot (1- x_n)$$

stb

Attractors

Lorenz attractor

Lorenz attractor is a strange attractor named after Edward Lorenz who described it in 1963.

For $\sigma = 10$, $\rho= 28$, $\beta = \frac 83$ and $x=0.1, y=0, z=0$ is the system described by the following three equations: $\begin{cases}\frac{dx}{dt} = \sigma (y-x)\\frac{dy}{dt} = x (\rho -z) -y\\frac{dz}{dt} = xy-\beta x\end{cases}$

stb

Thomas attractor

Thomas' cyclically symmetric attractor is a 3D strange attractor named after René Thomas.

For $\lambda = 0.19$ and $x=0.1, y=0, z=0$ is the system described by the following three equations: $\begin{cases}\frac {dx}{dt}= \sin(y) - \lambda x\\frac {dy}{dt}= \sin(z) - \lambda y\\frac {dz}{dt}= \sin(x) - \lambda z\end{cases}$

stb

Rössler attractor

Lorenz attractor is an attractor named after Otto Rössler who described it in 1976.

For $\alpha = 0.2$, $\beta= 0.2$, $\gamma= 14$ and $x=0.1, y=0, z=0$ is the system described by the following three equations: $\begin{cases}\frac {dx}{dt}=-y-z\\frac {dy}{dt}=x+\alpha y\\frac {dz}{dt}=\beta+z\cdot(x-\gamma)\end{cases}$

stb

Chen & Lu Chen attractor

The (Lu) Chen attractor is named after Jinhu Lu and Guanrong Chen.

For Chen attractor $\alpha = 40, \beta = 3, \gamma = 28$ and $x=-0.1, y=0.5, z=-0.6$ is the system described by the following three equations: $\begin{cases}\frac {dx}{dt}=\alpha \cdot (y - x)\\frac {dy}{dt}=(\gamma - \alpha) \cdot x - xz + \gamma y\\frac {dz}{dt}=xy - \beta z\end{cases}$

For Lu Chen attractor $\alpha = 34.5, \beta = 9, \gamma = 20, \omega = -10$ and $x = 0.1, y = 0.3, z = -0.6$ is the system described by the following three equations: $\begin{cases}\frac {dx}{dt}=\alpha \cdot (y - x)\\frac {dy}{dt}=x - xz + \gamma y + \omega\\frac {dz}{dt}=xy - \beta z\end{cases}$

Chen	Lu Chen

Rabinovich–Fabrikant attractor

The Rabinovich–Fabrikant equations are a set of three equations named after Mikhail Rabinovich and Anatoly Fabrikant, who described them in 1979.

For $\alpha = 1.1, \gamma = 0.87$ and $x=-1, y=0, z=0.5$ is the system described by the following three equations: $\begin{cases}\frac {dx}{dt}=y \cdot (z-1+x^2) + \gamma x\\frac {dy}{dt}=x\cdot (3z+1-x^2) + \gamma y\\frac {dz}{dt}=-2z\cdot (\alpha +xy)\end{cases}$

stb 1	stb 2

Hénon map

The Hénon map is a strange attractor named after Michel Hénon.

For $\alpha = 1.4, \beta= 0.3$ and $x=0.1, y=0.1$ is the system described by the following two equations: $\begin{cases}x_{n+1} = 1 - \alpha x_n^2 + y_n \y_{n+1} = \beta x_n\end{cases}$

stb

Fractals

Cantor set

• In ASCII
• In stb

A Cantor set is a fractal that is formed by removing the middle thirds of a line. It is named after Georg Cantor, who introduced it in 1883.

We get it by removing the interval $(\frac{1}3,\frac{2}3)$ of the content from the line $[0,1]$ . The open middle third of each of these remaining segments is deleted, and there are left these four segments: $[0,\frac{1}9] \cup [\frac{2}9,\frac{1}3] \cup [\frac{2}3,\frac{7}9] \cup [\frac{8}9,1]$.

The length of the Cantor line converges to $0$.

$$\frac{1}3 + \frac{2}9 + \frac{4}{27} + \frac{8}{81} + ... = \sum_{n=0}^{\infty} \frac{2^n}{3^{n+1}} = \frac{1}3 \sum_{n=0}^{\infty} \frac{2^n}{3^n} = \frac{1}3 (\frac{1}{1 - \frac{2}3 } ) = 1$$

The Cantor set gas a fractal dimension equal to $\frac{ln(2)}{ln(3)} \approx 0.6309...$.

ASCII	stb

Sierpiński carpet

• Available in ASCII
• Available in stb

A Sierpiński carpet is a fractal formed by recursively removing squares from a surface. It is named after Wacław Sierpiński, who described it in 1916 and it's a generalization of Cantor's set into two dimensions.

The area of a Sierpiński carpet converges to $0$.

$$S = 1 - \sum_{n=0}^\infty \frac{8^n}{9^{n+1}} \Longrightarrow a_1 = \frac{1}9, q = \frac{8}9 \Longrightarrow S = 1 - s = 1-\frac{\frac{1}9}{1-\frac{8}9} = 1 - 1 = 0$$

The Sierpiński carpet has a fractal dimension equal to $\frac{log(8)}{log(3)} \approx 1.8928...$ .

ASCII	stb

Sierpiński triangle

• Available in ASCII
• Available in stb

Mandelbrot set

• Available in ASCII
• Available in stb
• Available in SDL

ASCII	stb	SDL

Koch snowflake

• Available in ASCII
• Available in stb

Barnsley fern

• Available in ASCII
• Available in stb