Swirly Movement?

Are there any technique to create some swirly-flowy (dynamic) movement like, e.g., Flurries (Tuggummi), Arctic Rainbow (S_KuPeRS), etc.?
Do they model the equations right away from their mind (swirling imagery)?

learn your trigonometry, the window ranges from -1,1 both in x and y.
A lot could be learnt from reading and trying to understand the code in various presets.

Very simple.

init:
xa=ya;ya=-xa;

frame:
xt=xt+xa;
yt=yt+ya;
xa2=sin(xt)*.02;
ya2=sin(yt)*.02;
xz=1-sin(xa2);
yz=1-sin(ya2);

beat:
xa=rand(100)/1000-0.05;
ya=rand(100)/1000-0.05;

Pixel:
x=x*xz;x=x+sin(y+xt)*.05;
y=y*yz;y=y+sin(x+yt)*.05;