Homoclinic saddle-node bifurcations and subshifts in a three-dimensional flow

We study a two-parameter family of three-dimensional vector fields that are small perturbations of an integrable system possessing a line Gamma of deg enerate saddle points connected by a manifold of homoclinic loops. Under pe rturbation, this manifold splits and undergoes a quadratic homoclinic tange ncy. Perturbation methods followed by geometrical analyses reveal the prese nce of countably-infinite sets of homoclinic orbits to Gamma and a non-wand ering set topologically conjugate to a shift on two symbols (a Smale horses hoe). We use the symbolic description to identify and partially order bifur cation sequences in which the homoclinic orbits appear, and we formally der ive an explicit two-dimensional Poincare return map to further illustrate o ur results. The problem was motivated by the search for travelling 'structu res' such as fronts and domain walls in partial differential equations.