We address the stability of solitary pulses as well as some other traveling structures near the onset of spatiotemporal chaos in a two-species reaction-diffusion model describing the oxidation of CO on a Pt(110) surface in one spatial dimension. First, the boundary of the existence region of stable pulses is explored by means of numerical integration of the reaction-diffusion equations. The partial differential equations (PDEs) of the model are next reduced to a set of ordinary differential equations (ODEs) by the introduction of a moving frame and a detailed analysis of traveling wave solutions and their bifurcations is presented. The results are then compared to findings in numerical simulations and stability computations in the full PDE. The solutions of the ODE are organized around a codimension-2 global bifurcation from which two branches of homoclinic orbits corresponding to solitary pulse solutions in the PDE originate, This bifurcation mediates a change in the dynamics of the excitable medium, as seen in numerical simulations, from a regime dominated by stable pulses and wavetrains traveling with constant shape and speed to spatiotemporally chaotic dynamics. We also find a branch of heteroclinic orbits corresponding to fronts in the PDE. Even though these fronts are found to be unstable for the PDE, their spatial signature is frequently observed locally as part of the spatiotemporally chaotic profiles obtained by direct numerical simulation.