Opening a ring structure—the fold catastrophe

The catastrophe associated with the breaking of the H-H bond may be deseribed by a simple unfolding with one essential variable and one control parameter. The essential variable or behaviour coordinate lies along the Cj axis which is taken to be the z-axis. The control parameter corresponds to the location of the system along the reaction coordinate and this is given by R , the distance of the oxygen nucleus from the H-H midpoint. This so-called 

The analytical description of this fold catastrophe is obtained by replacing p(r) by a function of the single behaviour coordinate and parametrized by a single control parameter v, denoting displacements from the catastrophe point along the reaction coordinate. The topological behaviour of p(r) in the 

The behaviour variable contains implicitly the third derivative of p with respect to the actual coordinate z. The germ of this unfolding is the term (l/3)(, which is unstable as a critical point both the first, and second derivatives of this term are zero for ( = 0. If we. perturb it through the addition of a small term then its character is changed. This is seen by setting df/di = 0 which yields two roots for 

The complete description of the catastrophes involved in the dissociation of the water molecule must clearly involve two behaviour variables and at least two control parameters. We show in the following section that the most general description of the formation or destruction of a three-membered ring structure is described by such an unfolding, the elliptic umbilic. 

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