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

Abstract

We study a twoparameter family of threedimensional vector elds that are small perturbations of an integrable system possessing a line 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 analysis reveal the presence of countablyinnite sets of homoclinic orbits to and a nonwandering 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 twodimensional Poincare return map to further illustrate our results The problem was motivated by the search for traveling structures such as fronts and domain walls in partial dierential equations