  A model Hamiltonian for the reaction CH$_4^+ \rightarrow$ CH$_3^+$ + H,
parametrized to exhibit either early or late inner transition states, is
employed to investigate the dynamical characteristics of the roaming mechanism.
Tight/loose transition states and conventional/roaming reaction pathways are
identified in terms of time-invariant objects in phase space. These are
dividing surfaces associated with normally hyperbolic invariant manifolds
(NHIMs). For systems with two degrees of freedom NHIMS are unstable periodic
orbits which, in conjunction with their stable and unstable manifolds,
unambiguously define the (locally) non-recrossing dividing surfaces assumed in
statistical theories of reaction rates. By constructing periodic orbit
continuation/bifurcation diagrams for two values of the potential function
parameter corresponding to late and early transition states, respectively, and
using the total energy as another parameter, we dynamically assign different
regions of phase space to reactants and products as well as to conventional and
roaming reaction pathways. The classical dynamics of the system are
investigated by uniformly sampling trajectory initial conditions on the
dividing surfaces. Trajectories are classified into four different categories:
direct reactive and non reactive trajectories,which lead to the formation of
molecular and radical products respectively, and roaming reactive and non
reactive orbiting trajectories, which represent alternative pathways to form
molecular and radical products. By analysing gap time distributions at several
energies we demonstrate that the phase space structure of the roaming region,
which is strongly influenced by non-linear resonances between the two degrees
of freedom, results in nonexponential (nonstatistical) decay.
