Many physical systems lose or gain stability and pattern through bifurca- tion behavior. Extensive research of this behavior is carried out in many fields of science and engineering. The study of dynamic bifurcation be- havior, for example, has made clear the mechanism of dynamic instability and chaos. The group-theoretic bifurcation theory is an established means to deal with the formation and selection of patterns in association with symmetry-breaking bifurcation. Since all physical systems are "imperfect, " in that they inevitably involve some initial imperfections, the study of im- perfect bifurcation (bifurcation of imperfect systems) has drawn a keen mathematical interest to yield a series of important results, such as the universal unfolding. In structural mechanics, bifurcation behavior has been studied to model the buckling and failure of structural systems. The sharp reduction of the strength of structural systems by initial imperfections is formulated as im- perfection sensitivity laws. A series of statistical studies has been conducted to make clear the dependence of the strength of structures on the statis- tical variation of initial imperfections. A difficulty in these studies arises from the presence of a large number of initial imperfections. At this state, most of these studies are carried out based on the Monte Carlo simulation for a number of initial imperfections, or, on an imperfection sensitivity law against a single initial imperfection.
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