A general mathematical framework of bifurcation analysis that is to be employed throughout the book is presented. In particular, the Liapunov–Schmidt reduction is introduced as a tool to derive bifurcation equation. Perfect and imperfect bifurcation behaviors at simple critical points are investigated asymptotically in view of the leading terms of the power series expansion of this equation. This chapter lays a theoretical foundation of Chaps. 3 – 6 and is extended to a system with group symmetry in Chaps. 8 and 9.