The cusps of Zeeman's catastrophe machine

Abstract

W. BURKE asked me the following question. Zeeman’s catastrophe machine [ l] has four cusps in its bifurcation set; can one construct a catastrophe machine with just two cusps? This note shows that the answer to this question is no under rather broad hypotheses. Our answer to the question can be interpreted as a generalization of the four vertex theorem of elementary differential geometry[2]. The proof of our theorem provides yet another proof of the four vertex theorem, our proof being in the spirit of “catastrophe theory.” We shall first describe Zeeman’s catastrophe machine and state the theorem, then display the connection with differential geometry, and finally prove the theorem. Briefly recall the construction of Zeeman’s catastrophe machine. There is a disk D in the plane which is free to rotate around its center. Two springs S, and Sz are attached to one point p of the boundary of D. The spring S, has its other end attached at a fixed point in the plane. The other end u of Sz is free to be moved in the plane; see Fig. 1. The quantity which one measures with this machine is the stable equilibrium position(s) of the disk D as a function of the location of u ER*. If the position of D is measured by the angle 8 which p makes with the x-axis, then one determines the quantity 0(u). For some values of t’, there are two stable equilibria. The curve in R* across which the number of stable equilibria changes is called the bifurcation set of the machine. The bifurcation set of Zeeman’s machine has four cusps; see Fig. 2. An analysis of the catastrophe phenomenon which occurs across the bifurcation set can be made by introducing a potential function F(o, 0). The function F measures the potential energy of the system with the disk D held at position 0 and the end of the spring Sz held at u. The stable equilibria of the machine are given by pairs (u, 0) at which F has a local minimum when regarded as a function of 0 alone. The bifurcation set consists of u E R* at which the number of local minima change. In particular, a necessary condition for u to lie in the bifurcation set is that there exists a 8 for which (aF/ae)(u, 0) = (a’F/ae’)(u, e) = O.*