Rössler Attractor

The Rössler Attractor is what you get when you try to build the simplest possible chaotic system. In 1976 Otto Rössler looked at the Lorenz Attractor and asked whether chaos really needs two wings and two nonlinear terms. His answer was a system with only one nonlinear term:

With the standard values $a = 0.2$, $b = 0.2$ and $c = 5.7$, the orbit spirals outward in the $xy$-plane. Once it gets far enough from the center, the $z$ term switches on, lifts the orbit up into a sharp fold and drops it back near the middle of the spiral. Then the whole process starts over. That stretch-and-fold motion, a bit like kneading dough, is the basic recipe behind most chaotic systems.

The attractor is almost a flat band with one twist in it, so its fractal dimension is only just above two: estimates based on its Lyapunov exponents give about $2.01$. Cut through it and you would find a Cantor-like stack of very thin layers.

What you see here is a numerical solution with the classic fourth-order Runge-Kutta method. The parameter $c$ is the most fun to play with. For small values, around $c = 2.3$, the orbit closes into a simple loop. Increase it and the loop doubles to period 2, then 4, then 8, until it turns chaotic, the same period-doubling cascade you can see in the Logistic Map.