Title:

Bifurcations of transition states

A transition state for a Hamiltonian system is a closed, invariant, oriented, codimension2 submanifold of an energylevel that can be spanned by two compact codimension1 surfaces of unidirectional flux whose union, called a dividing surface, locally separates the energylevel into two components and has no local recrossings. For this to happen robustly to all smooth perturbations, the transition state must be normally hyperbolic. The dividing surface then has locally minimal geometric flux through it, giving a useful upper bound on the rate of transport in either direction. Transition states diffeomorphic to S2m−3 are known to exist for energies just above any index1 critical point of a Hamiltonian of m degrees of freedom, with dividing surfaces S2m−2. The question addressed here is what qualitative changes in the transition state, and consequently the dividing surface, may occur as the energy or other parameters are varied? We find that there is a class of systems for which the transition state becomes singular and then regains normal hyperbolicity with a change in diffeomorphism class. These are Morse bifurcations. Continuing the dividing surfaces and transition states through Morse bifurcations allows us to compute the flux for a larger range of energies. The effect of Morse bifurcations on the flux, as a function of energy, is considered and we find a loss of differentiability in the neighbourhood of the bifurcations. Various examples are considered. Firstly, some simple examples in which transition states connect or disconnect, and the dividing surface may become a torus or other. Then, we show that sequences of Morse bifurcations producing various interesting transition state and dividing surface are present in reacting systems, specifically bimolecular capture processes. We consider first planar reactions, for which the reduction of symmetries is easiest, and then also spatial reactions, where we find interesting Morse bifurcations involving both the attitude degrees of freedom and the angular momentum ones. In order to consider these examples, we present a method of constructing dividing surfaces spanning general transition states, and also a method to approximate normally hyperbolic submanifolds due to MacKay.
