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In chemical reactions, trajectories typically turn from reactants to productswhen crossing a dividing surface close to the normally hyperbolic invariantmanifold (NHIM) given by the intersection of the stable and unstable manifoldsof a rank-1 saddle. Trajectories started exactly on the NHIM in principle neverleave this manifold when propagated forward or backward in time. This stillholds for driven systems when the NHIM itself becomes time dependent. Weinvestigate the dynamics on the NHIM for a periodically driven model systemwith two degrees of freedom by numerically stabilizing the motion. UsingPoincar\'e surfaces of section we demonstrate the occurrence of structuralchanges of the dynamics, \emph{viz.}, bifurcations of periodic transition state(TS) trajectories when changing the amplitude and frequency of the externaldriving. In particular, periodic TS trajectories with the same period as theexternal driving but significantly different parameters---such as meanenergy---compared to the ordinary TS trajectory can be created in a saddle-nodebifurcation.

Dynamics and bifurcations on the normally hyperbolic invariant manifold of a periodically driven system with rank-1 saddle