BIFURCATIONS OF NORMALLY HYPERBOLIC INVARIANT MANIFOLDS IN ANALYTICALLY TRACTABLE MODELS AND CONSEQUENCES FOR REACTION DYNAMICS

Frederic A. L. Mauguiere*, Peter Collins, Gregory S. Ezra, Stephen Wiggins

*Corresponding author for this work

Research output: Contribution to journalArticle (Academic Journal)peer-review

11 Citations (Scopus)

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.

Original languageEnglish
Article number1330043
Number of pages20
JournalInternational Journal of Bifurcation and Chaos
Volume23
Issue number12
DOIs
Publication statusPublished - Dec 2013

Keywords

  • Normally hyperbolic invariant manifold
  • bifurcation
  • phase space dividing surface
  • reaction dynamics
  • transition state theory
  • TRANSITION-STATE THEORY
  • MONTE CARLO CALCULATIONS
  • PHASE-SPACE
  • UNIMOLECULAR DISSOCIATION
  • BIMOLECULAR COLLISIONS
  • ASYMPTOTIC STABILITY
  • TRAPPED TRAJECTORIES
  • REACTION PROBABILITY
  • HAMILTONIAN-SYSTEMS
  • CLASSICAL THEORIES

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