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

TL;DR

This paper studies how the dynamics on the normally hyperbolic invariant manifold (NHIM) in a periodically driven chemical system change through bifurcations, revealing new transition state trajectories as external parameters vary.

Contribution

It numerically analyzes the bifurcations of transition state trajectories on a time-dependent NHIM in a driven system, highlighting structural changes in the dynamics.

Findings

Bifurcations of periodic transition state trajectories occur with changing driving parameters.

New transition state trajectories with different energies can emerge via saddle-node bifurcations.

Structural changes in the dynamics are demonstrated using Poincaré sections.

Abstract

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.