The Lorenz attractor has a butterfly wing look. The dynamics spiral outwards, around first unstable fixed point, in the plane of one wing, before fly too close to the middle fixed point and getting deflected to the other wing. There the dynamics spiral outwards from the other "wing spot" fixed point.
Do things have to be that complicated? Could you spiral outward from (0,0,0) in the xy plane before being deflected upward by a repelling fixed point, and then falling back down to the xy plane, but close to the origin again? The Rössler Attractor does this, with a möbius strip twist to generate the chaos. The video has a cute, 3D printed model, differential equations, and animations. Every-thing you need to get inspired and try generating your own animation. (you will have to bring your own understanding of Xlib, or Canvas or whatever)
There doesn't seem to be anything here yet