Bifurcations of Normally Hyperbolic Invariant Manifolds and Consequences for Reaction Dynamics

TL;DR

This paper investigates how the breakdown of normal hyperbolicity affects reaction dynamics, including dividing surfaces and flux, using exactly solvable two-degree-of-freedom Hamiltonian models.

Contribution

It provides a detailed analysis of the effects of resonance-induced loss of hyperbolicity and topology changes in dividing surfaces within reaction dynamics models.

Findings

Resonances can cause loss of normal hyperbolicity or not.

Topology of dividing surface changes with resonances.

Flux varies continuously despite hyperbolicity loss.

Abstract

In this paper we study the breakdown of normal hyperbolicity and its consequences for reaction dynamics; in particular, the dividing surface, the flux through the dividing surface (DS), and the gap time distribution. Our approach is to study these questions using simple, two degree-of-freedom Hamiltonian models where calculations for the different geometrical and dynamical quantities can be carried out exactly. For our examples, we show that resonances within the normally hyperbolic invariant manifold may, or may not, lead to a `loss of normal hyperbolicity'. Moreover, we show that the onset of such resonances results in a change in topology of the dividing surface, but does not affect our ability to define a DS. The flux through the DS varies continuously with energy, even as the energy is varied in such a way that normal hyperbolicity is lost. For our examples the gap time distributions exhibit singularities at energies corresponding to the existence of homoclinic orbits in the DS, but these singularities are not associated with loss of normal hyperbolicity.