Analysis on evolution pattern of periodically distributed defects

A similar pattern is formed in various materials, when periodically distributed defects evolve. Mathematically, this pattern formation is understood as the consequence of symmetry breaking, while physically it is caused by interaction effect which vary depending on materials or defects. In examining the nature of the interaction effects, this paper analyzes the bifurcation induced growth of a periodic array of defects. With the aid of group-theoretic bifurcation analysis, it is clearly shown that when the uniform pattern (the evolution of all defects) is broken, only the alternate pattern (the evolution of every second defect) can take place for smaller defects, as often observed in nature. Therefore, two defects should be considered to examine a possible bifurcation of periodic defects. Furthermore, the conclusion obtained can be extended to explain the phenomena whereby every second, fourth, and then eighth defect continue to evolve, and whereby alternate bifurcation is repeated successively until the evolution is localized.