  In this article we present the influence of a Hamiltonian saddle-node
bifurcation on the high-dimensional phase space structures that mediate
reaction dynamics. To achieve this goal, we identify the phase space invariant
manifolds using Lagrangian descriptors, which is a trajectory-based diagnostic
suitable for the construction of a complete ``phase space tomography'' by means
of analyzing dynamics on low-dimensional slices. First, we build a Hamiltonian
system with one degree-of-freedom (DoF) that models reaction, and study the
effect of adding a parameter to the potential energy function that controls the
depth of the well. Then, we extend this framework to a saddle-node bifurcation
for a two DoF Hamiltonian, constructed by coupling a harmonic oscillator, i.e.
a bath mode, to the other reactive DoF in the system. For this problem, we
describe the phase space structures associated with the rank-1 saddle
equilibrium point in the bottleneck region, which is a Normally Hyperbolic
Invariant Manifold (NHIM) and its stable and unstable manifolds. Finally, we
address the qualitative changes in the reaction dynamics of the Hamiltonian
system due to changes in the well depth of the potential energy surface that
gives rise to the saddle-node bifurcation.
