CodingBobby | Shapes of Chaos
When studying fluid dynamics and heat transfer I learned about chaos and the attractors. My first encounter was with the Lorenz attractor, of course (read this blog post). The pure beauty of its unpredictable but organic shape hooked me from the first minute – I believe this feeling will stay with me forever.
So here is a small collection of rather simple but in a way artistic renderings of attractors I made for fun. Click on the images to find out more about them.
In a chaotic system, the trajectory moves around on the attractor as time goes on, but two nearby points separate exponentially so that eventually they are very far apart. Although their future is determined uniquely and precisely by the governing equations, very small differences in the starting point can make large differences in the future conditions. Although tomorrow’s weather depends on the conditions today, and the weather the day after tomorrow depends on the conditions tomorrow, small errors in measuring the current weather eventually grow until all hope of predictability is lost – the ‘butterfly effect’.
– Julien Clinton Sprott
A different kind of attactor emerge from non-differential systems that involve trigonometric functions like \(\sin\) and \(\cos\). They do not describe how a point moves along a smooth trajectory but rather where a given point jumps next. The result looks like smoke from a blown out candle that takes a strangely beautiful shape. You’ll find a gallery of interesting renders here.
Chaos in Phase Space
When not only one trajectory is shown (like for the attractors above), but many initial starting points across the 2D plane are fed into a differential system, we call the result the phase space of that respective system. Here, you can see some interesting patterns I’ve found.
Self-similar objects are called fractals. They offer literally infinite detail – you can zoom in as much as you want and you will always find more. Today’s fractal of the day by J.C. Sprott:
You can find some of my own graphics here.