  In chemical reactions, trajectories typically turn from reactants to products
when crossing a dividing surface close to the normally hyperbolic invariant
manifold (NHIM) given by the intersection of the stable and unstable manifolds
of a rank-1 saddle. Trajectories started exactly on the NHIM in principle never
leave this manifold when propagated forward or backward in time. This still
holds for driven systems when the NHIM itself becomes time dependent. We
investigate the dynamics on the NHIM for a periodically driven model system
with two degrees of freedom by numerically stabilizing the motion. Using
Poincar\'e surfaces of section we demonstrate the occurrence of structural
changes of the dynamics, \emph{viz.}, bifurcations of periodic transition state
(TS) trajectories when changing the amplitude and frequency of the external
driving. In particular, periodic TS trajectories with the same period as the
external driving but significantly different parameters---such as mean
energy---compared to the ordinary TS trajectory can be created in a saddle-node
bifurcation.
