  Reactivity boundaries that divide the destination and the origin of
trajectories are of crucial importance to reveal the mechanism of reactions. We
investigate whether such reactivity boundaries can be extracted for higher
index saddles in terms of a nonlinear canonical transformation successful for
index-one saddles by using a model system with an index-two saddle. It is found
that the true reactivity boundaries do not coincide with those extracted by the
transformation taking into account a nonlinearity in the region of the saddle
even for small perturbations, and the discrepancy is more pronounced for the
less repulsive direction of the index-two saddle system. The present result
indicates an importance of the global properties of the phase space to identify
the reactivity boundaries, relevant to the question of what reactant and
product are in phase space, for saddles with index more than one.
