  A transition state for a Hamiltonian system is a closed, invariant, oriented,
codimension-2 submanifold of an energy-level that can be spanned by two compact
codimension-1 surfaces of unidirectional flux whose union, called a dividing
surface, locally separates the energy-level 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 an upper bound on the rate of
transport in either direction. Transition states diffeomorphic to $\mathbb
S^{2m-3}$ are known to exist for energies just above any index-1 critical point
of a Hamiltonian of $m$ degrees of freedom, with dividing surfaces $\mathbb
S^{2m-2}$. 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. 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 how sequences of Morse bifurcations
producing various interesting forms of transition state and dividing surface
are present in reacting systems, by considering a hypothetical class of
bimolecular reactions in gas phase.
