The Study of Self-Organization Behavior on Sierpinski Gasket

Up till now, there is no report on the study of self-organization behavior on fractal networks. However, pattern formation on fractal networks is of great theoretical and experimental importance, for so many things in the world, incuding catalyst surface, should be viewed as fractal, and abundant patterns, e.g., turbulence, spiral waves and soliton waves have been observed in these systems. Using the reaction diffusion equation which describes wave propagating in excitable media, and adopting the fast simulation method provided by Barkley,the authols studied the pattern formation behavior on a kind of deterministic fractal : Sierpinski Gasket （SG） with the variation of the control parameters, turbulence, spiral waves and braid-like waves are observed. One sees that the defects of SG cannot hinder the formation of regular patterns. Compared to Euclidean space, however, one finds that the fractal structure can suppress the formation of more complex strctures, i.e., spiral waves or turbulence, which should result from the limitation of diffusion on fractal networks. It’s rather interesting that under certain parameter, a periodic excitation of pulse wave is observed, which collides with the braid-like wave and then disappeals. One should note that this phenomena is purely "structure indued". As a whole, this is the first report of the study on pattern formation in fractal media and it is demonstrated that fractal structure can lead to some new features which may calls the attention of scientists .