I think you might want to look into the Mandelbrot and Julia set, and things like Kochs Snowflake, as these are purely mathematical systems like in OP - this is

I think you might want to look into the Mandelbrot and Julia set, and things like Kochs Snowflake, as these are purely mathematical systems like in OP - this is one part of the book I didn't integrate as well, you might be satisfied by the book - but OP also lists some references on the symmetry of chaotic systems, maybe that's a better way forward!