Fractal and chaotic behavior of circular cellular automata

A new type of circular cellular automata (CCA) has been introduced. The evolutions of the CCA obtained by the clockwise, anticlockwise, and scanning line-by-line site sequence in the successively growing rings divided from a square lattice are studied. The evolution seems to form a twisty fishnet when the CCA are grown by the first two sequences, Sierpinski triangle gasket or the modulated ones are formed in the fourth quadrant of the CCA grown by the line scanning sequence. Fractal analysis is used to characterize the relationships between the pattern formed and the initial position of the growing ring and it is found that the pattern is very sensitive to the initial growth condition, showing the chaotic behavior.