The theory and application of bifurcation are presented in this chapter and chapter 5. This chapter describes the basic concepts of bifurcation in ordinary differential equations, Liapunov—Schmidt reduction (LS reduction for short), singularity theory and applications of all these theories. Chapter 5 introduces the centre manifold theorem and the normal form of vector fields. Chapter 6 presents the Hopf bifurcation and double zero eigenvalues. Chapter 7 explains the applications of the averaging method in bifurcation theory.