Dynamics and Bifurcations on the Normally Hyperbolic InvariantManifold of a Periodically Driven System with Rank-1 Saddle
2020
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 Poincare surfaces of
section, we demonstrate the occurrence of structural changes of the
dynamics, 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.
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