x = x*y+a*x-y (1)
y = x+y
as the parameter a varies from 0.01 to 0.35.
Instead this sequence of three images
shows the same chaotic attractor (represented in blue) for a=0.357057 .
This image is produced by iterating the formula starting from any point
in the range -4 <x,y<4 .
To understand how the attractor develops, we consider in the
same range a checkerboard red-white 200x200.
This is transformed using (1) applied once to the whole checkerboard in the image reported in red.
At each application of the formula (here you see only the first three iterations) the checkerboard is rotated,deformed and folded on itself.
The attractor is the final result of this procedure repeated many,many times.