Homoclinic bifurcations at the onset of pulse self-replication

We establish a series of properties of symmetric, N -pulse, homoclinic solutions of the reduced Gray– Scott system: u′′ = uv2, 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 (α,β)-plane that represent multi-pulse homoclinic orbits, where α and β are the central values of u(x) and v(x), respectively. We prove bounds for these solution branches, study their behavior as α →∞, and establish a series of geometric properties of these branches which are valid throughout the (α,β)-plane. We also establish qualitative properties of multi-pulse solutions and study how they bifurcate, i.e., how they change along the solution branches. © 2006 Elsevier Inc. All rights reserved. MSC: 35K45; 35K57; 35B25; 35B32; 35B35; 35B40; 34C30; 34C37; 92C15; 92E20