Homoclinic bifurcations at the onset of pulse self-replication

We establish a series of properties of symmetric, $N$-pulse, homoclinic solutions of the {\it reduced} Gray-Scott system $$ u'' = uv2, \qquad v''=v-uv2, $$
which play a pivotal role in questions concerning the existence and self-replication of pulse solutions of the full Gray-Scott model. Specifically, we establish the existence, and study properties, of solution branches in the $(\alpha,\beta)$-plane that represent multi-pulse homoclinic orbits, where $\aalpha$ and $\beta$ are the central values of $u(x)$ and $v(x)$, respectively. We prove bounds for these solution branches, study their behavior as $\alpha \to \infty$, and establish a series of geometric properties of these branches which are valid throughout the $(\alpha,\beta)$-plane. We also establish qualitative properties of multi-pulse solutions and study how they bifurcate, {\it i.e.}, how they change along the solution branches.