Bifurcations of transition states

摘要: A transition state for a Hamiltonian system is closed, invariant, oriented, codimension-2 submanifold of an energy-level that can be spanned by two compact codimension-1 surfaces unidirectional flux whose union, called dividing surface, locally separates the into components and has no local recrossings. For this to happen robustly all smooth perturbations, must normally hyperbolic. The surface then minimal geometric through it, giving useful upper bound on rate transport in either direction. Transition states diffeomorphic S2m−3 are known exist energies just above any index-1 critical point m degrees freedom, with S2m−2. question addressed here what qualitative changes state, consequently may occur as energy or other parameters varied? We find there class systems which becomes singular regains normal hyperbolicity change diffeomorphism class. These Morse bifurcations. Continuing bifurcations allows us compute larger range energies. effect flux, function energy, considered we loss differentiability neighbourhood bifurcations. Various examples considered. Firstly, some simple connect disconnect, become torus other. Then, show sequences producing various interesting present reacting systems, specifically bimolecular capture processes. consider first planar reactions, reduction symmetries easiest, also spatial where involving both attitude freedom angular momentum ones. In order these examples, method constructing spanning general states, approximate hyperbolic submanifolds due MacKay.