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In this paper we study the breakdown of normal hyperbolicity and itsconsequences for reaction dynamics; in particular, the dividing surface, theflux through the dividing surface (DS), and the gap time distribution. Ourapproach is to study these questions using simple, two degree-of-freedomHamiltonian models where calculations for the different geometrical anddynamical quantities can be carried out exactly. For our examples, we show thatresonances within the normally hyperbolic invariant manifold may, or may not,lead to a `loss of normal hyperbolicity'. Moreover, we show that the onset ofsuch resonances results in a change in topology of the dividing surface, butdoes not affect our ability to define a DS. The flux through the DS variescontinuously with energy, even as the energy is varied in such a way thatnormal hyperbolicity is lost. For our examples the gap time distributionsexhibit singularities at energies corresponding to the existence of homoclinicorbits in the DS, but these singularities are not associated with loss ofnormal hyperbolicity.

Bifurcations of Normally Hyperbolic Invariant Manifolds and Consequences for Reaction Dynamics