Homoclinic Saddle‐Node Bifurcations and Subshifts in a Three‐Dimensional Flow

Abstract.

We study a two‐parameter family of three‐dimensional vector fields that are small perturbations of an integrable system possessing a line Γ of degenerate saddle points connected by a manifold of homoclinic loops. Under perturbation, this manifold splits and undergoes a quadratic homoclinic tangency. Perturbation methods followed by geometrical analyses reveal the presence of countably‐infinite sets of homoclinic orbits to Γ and a non‐wandering set topologically conjugate to a shift on two symbols (a Smale horseshoe). We use the symbolic description to identify and partially order bifurcation sequences in which the homoclinic orbits appear, and we formally derive an explicit two‐dimensional Poincaré return map to further illustrate our results. The problem was motivated by the search for travelling ‘structures’ such as fronts and domain walls in partial differential equations.