Classification by Codimension

Abstract

The main purpose of this chapter is to classify all bifurcation problems (in one state variable) of codimension three or less. We find that there are eleven such singularities, which we call the elementary bifurcation problems. In the course of the chapter, we tabulate the following data for each of these eleven singularities:

• (i) Normal form (Table 2.1).

• (ii) Algebraic data (i.e., J(h), RT(h), T(h), a complement to T(h), codimension) (Table 2.2).

• (iii) Solution of the recognition problem for normal forms (Table 2.3).

• (iv) Universal unfolding (Table 3.1).

• (v) Solution of the recognition problem for universal unfoldings (Table 3.2).

• (vi) Equations for the bifurcation, hysteresis, and double limit point varieties (Table 4.1).

• (vii) Graphs of the persistent perturbed bifurcation diagrams (Figures 4.1-4.3).

Thus the chapter should also be useful as a compact reference.