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 qu
estions 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.
We investigate the fragmentation dynamics of an atomic chain under tensile stress. We have classified the location, stability type (indices) and energy of all equilibria for the general $n$-particle chain, and have highlighted the importance of saddl
e points with index $> 1$. We show that for an $n=2$-particle chain under tensile stress the index 2 saddle plays a central role in organizing the dynamics. We apply normal form theory to analyze phase space structure and dynamics in a neighborhood of the index 2 saddle. We define a phase dividing surface (DS) that enables us to classify trajectories passing through a neighborhood of the saddle point using the values of the integrals associated with the normal form. We also generalize our definition of the dividing surface and define an emph{extended dividing surface} (EDS), which is used to sample and classify all trajectories that pass through a phase space neighborhood of the index 2 saddle at total energies less than that of the saddle. Classical trajectory simulations are used to study single versus double bond breakage for the $n=2$ chain under tension. Initial conditions for trajectories are obtained by sampling the EDS at constant energy. We sample trajectories at fixed energies both above and below the energy of the saddle. The fate of trajectories (single versus double bond breakage) is explored as a function of the location of the initial condition on the EDS, and a connection made to the work of Chesnavich on collision-induced dissociation. A significant finding is that we can readily identify trajectories that exhibit bond emph{healing}. Such trajectories pass outside the nominal (index 1) transition state for single bond dissociation, but return to the potential well region, possibly several times, before ultimately dissociating.
