Hamiltonian pitchfork bifurcation in transition across index-1 saddles

TL;DR

This paper investigates how parameter changes in a 2D potential energy surface induce bifurcations affecting phase space structures and reaction dynamics, with applications to chemical reactions like isomerization and dissociation.

Contribution

It introduces a quartic Hamiltonian exhibiting pitchfork bifurcation, derives bifurcation criteria, and analyzes phase space structures and reaction dynamics dependence on energy and coupling.

Findings

Bifurcation criteria for the Hamiltonian are derived.

Phase space structures such as unstable periodic orbits are characterized.

Reaction dynamics depend on energy and coupling strength, affecting flux and gap times.

Abstract

We study the effect of changes in the parameters of a two-dimensional potential energy surface on the phase space structures relevant for chemical reaction dynamics. The changes in the potential energy are representative of chemical reactions such as isomerization between two structural conformations or dissociation of a molecule with an intermediate. We present a two degrees of freedom quartic Hamiltonian that shows pitchfork bifurcation when the parameters are varied and we derive the bifurcation criteria relating the parameters. Next, we describe the phase space structures - unstable periodic orbits and their associated invariant manifolds, and phase space dividing surfaces - for the systems that can show trajectories undergo reaction defined as crossing of a potential energy barrier. Finally, we quantify the reaction dynamics for these systems by obtaining the directional flux and gap time distribution to illustrate the dependence on total energy and the coupling strength between the two degrees of freedom.